Artin Transfer Patterns on Descendant Trees of Finite p-Groups ()
Received 25 November 2015; accepted 25 January 2016; published 29 January 2016
![]()
1. Introduction
P 1.1. In the mathematical field of group theory, an Artin transfer is a certain homomorphism from an arbitrary finite or infinite group to the commutator quotient group of a subgroup of finite index.
Originally, such transfer mappings arose as group theoretic counterparts of class extension homomorphisms of abelian extensions of algebraic number fields by applying Artin’s reciprocity isomorphism ([1] , §4, Allgemeines Reziprozitätsgesetz, p. 361) to ideal class groups and analyzing the resulting homomorphisms between quotients of Galois groups ([2] , §2, p. 50).
However, independently of number theoretic applications, a natural partial order on the kernels and targets of Artin transfers, has recently been found to be compatible with parent-child relations between finite p-groups, where p denotes a prime number. Such ancestor-descendant relations can be visualized conveniently in des- cendant trees ([3] , §4, pp. 163-164).
Consequently, Artin transfers provide valuable information for classifying finite p-groups by kernel-target patterns and for searching and identifying particular groups in descendant trees by looking for patterns defined by kernels and targets of Artin transfers. These strategies of pattern recognition are useful not only in purely group theoretic context, but also, most importantly, for applications in algebraic number theory concerning Galois groups of higher p-class fields and Hilbert p-class field towers. The reason is that the unramified extensions of a base field contain information in the shape of capitulation patterns and class group structures, and these arithmetic invariants can be translated into group theoretic data on transfer kernels and targets by means of Artin’s reciprocity law of class field theory. The natural partial order on Artin patterns admits termination criteria for a search through a descendant tree with the aid of recursive executions of the p-group generation algorithm by Newman [4] and O’Brien [5] .
P 1.2. The organization of this article is as follows. The detailed theory of the transfer will be developed in §§ 2 and 3, followed by computational implementations in § 4. It is our intention to present more than the least common multiple of the original papers by Schur [6] and Artin [2] and the relevant sections of the text books by Hall [7] , Huppert [8] , Gorenstein [9] , Aschbacher [10] , Doerk and Hawkes [11] , Smith and Tabachnikova [12] , and Isaacs [13] .
However, we shall not touch upon fusion and focal subgroups, which form the primary goal of the mentioned authors, except Artin. Our focus will rather be on a sound foundation of Artin patterns, consisting of families of transfer kernels and targets, and their stabilization, resp. polarization, in descendant trees of finite p-groups. These phenomena arise from a natural partial order on Artin patterns which is compatible with ancestor- descendant relations in trees, and is established in its most general form in §§5 and 6.
Since our endeavour is to give the most general view of each partial result, we came to the conviction that categories, functors and natural transformations are the adequate tools for expressing the appropriate range of validity for the facts connected with the partial order relation on Artin patterns. Inspired by Bourbaki’s method of exposition [14] , Appendix on induced homomorphisms, which is separated to avoid a disruption of the flow of exposition, goes down to the origins exploiting set theoretic facts concerning direct images and inverse pre-images of mappings which are crucial for explaining the natural partial order of Artin patterns.
2. Transversals and Their Permutations
2.1. Transversals of a Subgroup
Let G be a group and
be a subgroup of finite index
.
Definition 2.1. See also ([6] , p. 1013); ([7] , (1.5.1), p. 11); ([8] , Satz 2.5, p. 5).
1). A left transversal of H in G is an ordered system
of representatives for the left cosets of H in
G such that
is a disjoint union.
2). Similarly, a right transversal of H in G is an ordered system
of representatives for the right
cosets of H in G such that
is a disjoint union.
Remark 2.1. For any transversal of H in G, there exists a unique subscript
such that
, resp.
. The element
, resp.
, which represents the principal coset (i.e., the subgroup H itself) may be replaced by the neutral element 1.
Lemma 2.1. See also ([6] , p. 1015); ([7] , (1.5.2), p. 11); ([8] , Satz 2.6, p. 6).
1). If G is non-abelian and H is not a normal subgroup of G, then we can only say that the inverse elements
of a left transversal
form a right transversal of H in G.
2). However, if
is a normal subgroup of G, then any left transversal is also a right transversal of H in G.
Proof. 1). Since the mapping
,
is an involution, that is a bijection which is its own inverse, we see that
implies
.
2). For a normal subgroup
, we have
for each
.
![]()
Let
be a group homomorphism and
be a left transversal of a subgroup H in G with finite index
. We must check whether the image of this transversal under the homomorphism is a transversal again.
Proposition 2.1. The following two conditions are equivalent.
1).
is a left transversal of the subgroup
in the image
with finite
index
.
2).
.
We emphasize this important equivalence in a formula:
(2.1)
Proof. By assumption, we have the disjoint left coset decomposition
which comprises two
statements simultaneously.
Firstly, the group
is a union of cosets,
and secondly, any two distinct cosets have an empty intersection
, for
.
Due to the properties of the set mapping associated with
, the homomorphism
maps the union to another union
![]()
but weakens the equality for the intersection to a trivial inclusion
![]()
To show that the images of the cosets remain disjoint we need the property
of the homo- morphism
.
Suppose that
for some
,
then we have
for certain elements
.
Multiplying by
from the left and by
from the right, we obtain
![]()
Since
, this implies
, resp.
, and thus
. (This part of the proof is also covered by ([13] , Thm. X. 21, p. 340) and, in the context of normal subgroups instead of homomorphisms, by ([7] , Thm. 2.3.4, p. 29) and ([8] , Satz 3.10, p. 16))
Conversely, we use contraposition.
If the kernel
of
is not contained in the subgroup H, then there exists an element
such that
.
But then the homomorphism
maps the disjoint cosets ![]()
to equal cosets
.
□
2.2. Permutation Representation
P 2.1. Suppose
is a left transversal of a subgroup
of finite index
in a group G. A fixed element
gives rise to a unique permutation
of the left cosets of H in G by left multiplication such that
(2.2)
for each
.
Similarly, if
is a right transversal of H in G, then a fixed element
gives rise to a unique permutation
of the right cosets of H in G by right multiplication such that
(2.3)
for each
.
The elements
, resp.
,
, of the subgroup H are called the monomials associated with x with respect to
, resp.
.
Definition 2.2 See also ([8] , Hauptsatz 6.2, p. 28).
The mapping
, resp.
, is called the permutation representation of G in
with respect to
, resp.
.
Lemma 2.2. For the special right transversal
associated to the left transversal
, we have the following relations between the monomials and permutations corresponding to an element
:
(2.4)
Proof. For the right transversal
, we have
, for each
.
On the other hand, for the left transversal
, we have
, for each
.
This relation simultaneously shows that, for any
, the permutation representations and the associated monomials are connected by
![]()
for each
. □
3. Artin Transfer
Let G be a group and
be a subgroup of finite index
. Assume that
, resp.
, is a left, resp. right, transversal of H in G with associated permutation representation
,
, resp.
, such that
, resp.
, for
.
Definition 3.3. See also ([6] , p. 1014); ([2] , §2, p. 50); ([7] , (14.2.2-4), p. 202); ([8] , p. 413); ([9] , p. 248); ([10] , p. 197); ([11] , Dfn.(17.1), p. 60); ([12] , p. 154); ([13] , p. 149); ([15] , p. 2).
The Artin transfer
from G to the abelianization
of H with respect to
, resp.
, is defined by
(3.1)
resp.
(3.2)
for
.
Remark 3.1. I.M. Isaacs [13] , p. 149 calls the mapping
,
, resp.
,
the pre-transfer from G to H. The pre-transfer can be composed with a homomorphism
from H into
an abelian group A to define a more general version of the transfer
,
, resp.
, from G to A via
, which occurs in the book by D. Gorenstein ([9] , p. 248). Taking the
natural epimorphism
,
, yields the Definition 3.3 of the Artin transfer
in its original form by I. Schur ([6] , p. 1014) and by E. Artin ([2] , §2, p. 50), which has also been dubbed Verlagerung by H. Hasse ([16] , §27.4, pp. 170-171). Note that, in general, the pre-transfer is neither independent of the transversal nor a group homomorphism.
3.1. Independence of the Transversal
Assume that
is another left transversal of H in G such that
.
Proposition 3.1. See also ([6] , p. 1014); ([7] , Thm. 14.2.1, p. 202); ([8] , Hilfssatz 1.5, p. 414); ([9] , Thm. 3.2, p. 246); ([10] , (37.1), p.198); ([11] , Thm.(17.2), p.61); ([12] , p.154); ([13] , Thm.5.1, p.149); ([15] , Prop.2, p. 2).
The Artin transfers with respect to (g) and
coincide,
.
Proof. There exists a unique permutation
such that
, for all
. Consequently,
, resp.
with
, for all
. For a fixed element
, there exists a unique permutation
such that we have
,
for all
. Therefore, the permutation representation of G with respect to
is given by
, resp.
, for
. Furthermore, for the connection between the elements
and
, we obtain
![]()
for all
. Finally, due to the commutativity of the quotient group
and the fact that
and
are permutations, the Artin transfer turns out to be independent of the left transversal:
![]()
as prescribed in Definition 3.1, Equation (3.1). □
It is clear that a similar proof shows that the Artin transfer is independent of the choice between two different right transversals. It remains to show that the Artin transfer with respect to a right transversal coincides with the Artin transfer with respect to a left transversal.
For this purpose, we select the special right transversal
associated to the left transversal
, as explained in Lemma 2.1 and Lemma 2.2.
Proposition 3.2. The Artin transfers with respect to
and
coincide,
.
Proof. Using (2.4) in Lemma 2.2 and the commutativity of
, we consider the expression
![]()
The last step is justified by the fact that the Artin transfer is a homomorphism. This will be shown in the following subsection 3.2. □
3.2. Artin Transfers as Homomorphisms
Let
be a left transversal of H in G.
Theorem 3.1. See also ([6] , p. 1014); ([7] , Thm. 14.2.1, p. 202); ([8] , Hauptsatz 1.4, p. 413); ([9] , Thm. 3.2, p. 246); ([10] , (37.2), p.198); ([11] , Thm.(17.2), p.61); ([12] , p. 155); ([13] , Thm.5.2, p. 150); ([15] , Prop.1, p. 2).
The Artin transfer
,
and the permutation representation
are group homomorphisms:
(3.3)
Proof. Let
be two elements with transfer images
and
. Since
is abelian and
is a permutation, we can change the order of the factors in the following product:
![]()
This relation simultaneously shows that the Artin transfer
and the permutation representation
are homomorphisms, since
and
, in a covariant way. □
3.3. Monomial Representation
Let
, resp.
, be a left, resp. right, transversal of a subgroup H in a group G. Using the monomials
, resp.
, associated with an element
according to Equation (2.2), resp. (2.3), we define the following maps.
Definition 3.2. The mapping
, respectively
, is called the monomial representation of G in
with respect to
, resp.
.
P 3.1. It is illuminating to restate the homomorphism property of the Artin transfer in terms of the monomial
representation. The images of the factors
are given by
and
. In the proof of Theorem 3.1, the image of the product
turned out to be
, which is a very peculiar law of com- position discussed in more detail in the sequel.
The law reminds of the crossed homomorphisms
in the first cohomology group
of a G-module M, which have the property
, for
.
These peculiar structures can also be interpreted by endowing the cartesian product
with a special law of composition known as the wreath product
of the groups H and
with respect to the set
.
Definition 3.3. For
, the wreath product of the associated monomials and permutations is given by
(3.4)
Theorem 3.2. See also ([7] , Thm.14.1, p. 200); ([8] , Hauptsatz 1.4, p. 413).
This law of composition on
causes the monomial representation ![]()
also to be a homomorphism. In fact, it is a faithful representation, that is an injective homomorphism, also called a monomorphism or embedding, in contrast to the permutation representation.
Proof. The homomorphism property has been shown above already. For a homomorphism to be injective, it suffices to show the triviality of its kernel. The neutral element of the group
endowed with the wreath product is given by
, where the last 1 means the identity permutation. If
, for some xÎG, then
and consequently
, for all
. Finally, an application of the inverse inner automorphism with
yields
, as required for injectivity.
The permutation representation cannot be injective if G is infinite or at least of an order bigger than
, the factorial of n. □
Remark 3.2. Formula (3.4) is an example for the left-sided variant of the wreath product on
. However, we point out that the wreath product with respect to a right transversal
of H in G appears in its right-sided variant
(3.5)
which implies that the permutation representation
,
is a homomorphism with respect to the opposite law of composition
on
, in a contravariant manner.
It can be shown that the left-sided and the right-sided variant of the wreath product lead to isomorphic group structures on
.
A related viewpoint is taken by M. Hall ([7] , p. 200), who uses the multiplication of monomial matrices to describe the wreath product. Such a matrix can be represented in the form
as the product of an invertible diagonal matrix over the group ring K[H], where K denotes a field, and the permutation matrix
associated with the permutation
. Multiplying two such monomial matrices yields a law of composition identical to the wreath product in the right-sided variant,
![]()
Whereas B. Huppert ([8] , p. 413) uses the monomial representation for defining the Artin transfer by composition with the unsigned determinant, we prefer giving the immediate Definition 3.3 and merely illustrating the homomorphism property of the Artin transfer with the aid of the monomial representation.
3.4. Composition of Artin Transfers
Let G be a group with nested subgroups
such that the indices
,
and
are finite.
Theorem 3.3. See also ([7] , Thm.14.2.1, p. 202); ([8] , Satz 1.6, p. 415); ([11] , Lem.(17.3), p. 61); ([13] , Thm.10.8, p. 301); ([15] , Prop.3, p. 3).
Then the Artin transfer
is the compositum of the induced transfer
(in the sense of Corollary 7.1 or Corollary 7.3 in the Appendix) and the Artin transfer
, i.e.,
(3.6)
This can be seen in the following manner.
Proof. If
is a left transversal of H in G and
is a left transversal of K in H, that is
and
, then
is a disjoint left coset decomposition of G with
respect to K. See also ([7] , Thm.1.5.3, p. 12); ([8] , Satz 2.6, p. 6). Given two elements
and
, there exist unique permutations
, and
, such that the associated monomials are given by
, for each
, and
, for each
.
Then, using Corollary 7.3, we have
, and
.
For each pair of subscripts
and
, we put
and obtain
![]()
resp.
. Thus, the image of x under the Artin transfer
is given by
![]()
□
3.5. Wreath Product of Sm and Sn
P 3.2. Motivated by the proof of Theorem 3.3, we want to emphasize the structural peculiarity of the monomial representation
![]()
which corresponds to the compositum of Artin transfers, defining
![]()
for a permutation
, and using the symbolic notation
for all pairs of subscripts
,
.
The preceding proof has shown that
. Therefore, the action of the permutation
on the set
is given by
. The action on the second component j depends
on the first component i (via the permutation
), whereas the action on the first component i is independent of the second component j. Therefore, the permutation
can be identified with the multiplet
, which will be written in twisted form in the sequel.
The permutations
, which arise as second components of the monomial representation
![]()
are of a very special kind. They belong to the stabilizer of the natural equipartition of the set [1, n] × [1, m] into the n rows of the corresponding matrix (rectangular array). Using the peculiarities of the composition of Artin transfers in the previous section, we show that this stabilizer is isomorphic to the wreath product
of the symmetric groups
and
with respect to
, whose underlying set
is endowed with the following law of composition in the left-sided variant.
(3.7)
for all
.
This law reminds of the chain rule
for the Fréchet derivative in
xÎE of the compositum of differentiable functions
and
between complete normed spaces.
The above considerations establish a third representation, the stabilizer representation,
![]()
of the group G in the wreath product
, similar to the permutation representation and the monomial representation. As opposed to the latter, the stabilizer representation cannot be injective, in general. For instance, certainly not, if G is infinite.
Formula (3.7) proves the following statement.
Theorem 3.4. The stabilizer representation
of the group G in
the wreath product
of symmetric groups is a group homomorphism.
3.6. Cycle Decomposition
Let
be a left transversal of a subgroup
of finite index
in a group G. Suppose the element
gives rise to the permutation
of the left cosets of H in G such that
, resp.
, for each
.
Theorem 3.5. See also ([2] , §2, p. 50); ([16] , §27.4, p. 170); ([8] , Hilfssatz 1.7, p. 415); ([9] , Thm.3.3, p. 249); ([10] , (37.3), p. 198); ([12] , p. 154); ([13] , Lem.5.5, p. 153); ([15] , p. 5).
If the permutation
has the decomposition
into pairwise disjoint (and thus commuting)
cycles
of lengths
, which is unique up to the ordering of the cycles, more explicitly, if
(3.8)
for
, and
, then the image of
under the Artin transfer
is given by
(3.9)
Proof. The reason for this fact is that we obtain another left transversal of H in G by putting
for
and
, since
(3.10)
is a disjoint decomposition of G into left cosets of H.
Let us fix a value of
. For
, we have
![]()
However, for
, we obtain
![]()
Consequently,
![]()
□
P 3.3. The cycle decomposition corresponds to a double coset decomposition
of the group
G modulo the cyclic group
and modulo the subgroup H. It was actually this cycle decomposition form of the transfer homomorphism which was given by E. Artin in his original 1929 paper ([2] , §2, p. 50).
3.7. Transfer to a Normal Subgroup
P 3.4. Now let
be a normal subgroup of finite index
in a group G. Then we have
, for all
, and there exists the quotient group
of order n. For an element
, we let
denote the order of the coset
in
, and we let
be a left transversal of the subgroup
in G, where
.
Theorem 3.6. See also ([16] , §27.4, VII, p. 171).
Then the image of
under the Artin transfer
is given by
(3.11)
Proof.
is a cyclic subgroup of order f in
, and a left transversal
of the subgroup
in G, where
and
is the corresponding disjoint left coset decomposition,
can be refined to a left transversal
with disjoint left coset decomposition
(3.12)
of H in G. Hence, the formula for the image of x under the Artin transfer
in the previous section takes the particular shape
![]()
with exponent f independent of j. □
Corollary 3.1. See also ([13] , Lem.10.6, p. 300) for a special case.
In particular, the inner transfer of an element
is given as a symbolic power
(3.13)
with the trace element
(3.14)
of H in G as symbolic exponent.
The other extreme is the outer transfer of an element
which generates G modulo H, that is
. It is simply an nth power
(3.15)
Proof. The inner transfer of an element
, whose coset
is the principal set in
of order
, is given as the symbolic power
![]()
with the trace element
![]()
of H in G as symbolic exponent.
The outer transfer of an element
which generates G modulo H, that is
, whose coset
is generator of
with order
, is given as the nth power
.
□
P 3.5. Transfers to normal subgroups will be the most important cases in the sequel, since the central concept of this article, the Artin pattern, which endows descendant trees with additional structure, consists of targets and kernels (§5) of Artin transfers from a group G to intermediate groups
between G and its com- mutator subgroup
. For these intermediate groups we have the following lemma.
Lemma 3.1. All subgroups
of a group G which contain the commutator subgroup
are normal subgroups
.
Proof. Let
. If H were not a normal subgroup of G, then we had
for some element
. This would imply the existence of elements
and
such that
, and consequently the commutator
would be an element in
in contradiction to
. □
Explicit implementations of Artin transfers in the simplest situations are presented in the following section.
4. Computational Implementation
4.1. Abelianization of Type (p, p)
P 4.1. Let G be a pro-p group with abelianization
of elementary abelian type
. Then G has
maximal subgroups
of index
. In this particular case, the Frattini
subgroup
, which is defined as the intersection of all maximal subgroups, coincides with the
commutator subgroup
, since the latter contains all pth powers
, and thus we have
.
For each
, let
be the Artin transfer homomorphism from G to the abelianization of
. According to Burnside's basis theorem, the group G has generator rank
and can therefore be generated as
by two elements
such that
. For each of the normal subgroups
, we need a generator
with respect to
, and a generator
of a transversal![]()
such that
and
.
A convenient selection is given by
(4.1)
Then, for each
, it is possible to implement the inner transfer by
(4.2)
according to Equation (3.13) of Corollary 3.1, which can also be expressed by a product of two pth powers,
(4.3)
and to implement the outer transfer as a complete pth power by
(4.4)
according to Equation (3.15) of Corollary 3.1. The reason is that
and
in the quotient group
.
It should be pointed out that the complete specification of the Artin transfers
also requires explicit knowledge of the derived subgroups
. Since
is a normal subgroup of index p in
, a certain general reduction is possible by
([17] , Lem.2.1, p. 52), but an explicit pro-p pre- sentation of G must be known for determining generators of
, whence
(4.5)
4.2. Abelianization of Type (p2, p)
P 4.2. Let G be a pro-p group with abelianization
of non-elementary abelian type
. Then G has
maximal subgroups
of index
and
subgroups ![]()
of index
.
Figure 1 visualizes this smallest non-trivial example of a multi-layered abelianization
([18] , Dfn.3.1- 3, p. 288).
For each
, let
, resp.
, be the Artin transfer homo- morphism from G to the abelianization of
, resp.
. Burnside’s basis theorem asserts that the group G has generator rank
and can therefore be generated as
by two elements
such that
.
We begin by considering the first layer of subgroups. For each of the normal subgroups
, we select a generator
(4.6)
These are the cases where the factor group
is cyclic of order
. However, for the distinguished maximal subgroup
, for which the factor group
is bicyclic of type
, we need two generators
(4.7)
Further, a generator
of a transversal must be given such that
, for each
. It is convenient to define
(4.8)
Then, for each
, we have the inner transfer
(4.9)
which equals
, and the outer transfer
(4.10)
since
and
.
Now we continue by considering the second layer of subgroups. For each of the normal subgroups
, we select a generator
(4.11)
such that
. Among these subgroups, the Frattini subgroup
is par- ticularly distinguished. A uniform way of defining generators
of a transversal such that
, is to set
(4.12)
Since
, but on the other hand
and
, for
, with the single exception that
, we obtain the following expressions for the inner transfer
(4.13)
and for the outer transfer
(4.14)
exceptionally
(4.15)
and
(4.16)
for
. Again, it should be emphasized that the structure of the derived subgroups
and
must be known explicitly to specify the action of the Artin transfers completely.
5. Transfer Kernels and Targets
P 5.1. After our thorough treatment of the general theory of Artin transfers in §§2 and 3, and their computational implementation for some simple cases in §4, we are now in the position to introduce Artin transfer patterns, which form the central concept of this article. They provide an incredibly powerful tool for classifying finite and infinite pro-p groups and for identifying a finite p-group G with sufficiently many assigned components of its Artin pattern by the strategy of pattern recognition. This is done in a search through the descendant tree with root
by means of recursive applications of the p-group generation algorithm by Newman [4] and O’Brien [5] .
An Artin transfer pattern consists of two families of transfer targets, resp. kernels, which are also called multiplets, whereas their individual components are referred to as singulets.
5.1. Singulets of Transfer Targets
Theorem 5.1. Let G and T be groups. Suppose that
is the image of G under a homomorphism
, and
is the image of an arbitrary subgroup
. Then the following claims hold without any further necessary assumptions.
1) The commutator subgroup of V is the image of the commutator subgroup of U, that is
(5.1)
2) The restriction
is an epimorphism which induces a unique epimorphism
(5.2)
Thus, the abelianization of V,
(5.3)
is an epimorphic image of the abelianization of U, namely the quotient of
by the kernel of
, which is given by
(5.4)
3) Moreover, the map
is an isomorphism, and the quotients
are isomorphic, if and only if
(5.5)
See Figure 2 for a visualization of this situation.
Proof. The statements can be seen in the following manner. The image of the commutator subgroup is given by
![]()
The homomorphism
can be restricted to an epimorphism
. According to Theorem 7.1, in particular, by the Formulas (7.5) and (7.4) in the appendix, the condition
implies the existence of a uniquely determined epimorphism
such that
. The Isomor- phism Theorem in Formula (7.7) in the appendix shows that
. Furthermore, by the Formulas (7.4) and (7.1), the kernel of
is given explicitly by
![]()
Thus,
is an isomorphism if and only if
. □
P 5.2. Functor of derived quotients. In analogy to section §7.6 in the appendix, a covariant functor
can be used to map a morphism
of one category to an induced morphism
of another category.
In the present situation, we denote by
the category of groups and we define the domain of the functor F as the following category
. The objects of the category are pairs
consisting of a group G and a subgroup
,
(5.6)
For two objects
, the set of morphisms
consists of epimor- phisms
such that
and
, briefly written as arrows
,
(5.7)
The functor
from this category
to the category
of abelian groups maps a pair
to the commutator quotient group
of the subgroup
, and
it maps a morphism
to the induced epimorphism
of the restriction
,
(5.8)
Existence and uniqueness of
have been proved in Theorem 5.1 under the assumption that
, which is satisfied according to the definition of the arrow
and automatically implies
.
![]()
Figure 2. Induced homomorphism of derived quotients.
Definition 5.1. Due to the results in Theorem 5.1, it makes sense to define a pre-order of transfer targets on the image
of the functor F in the object class
of the category
of abelian groups in the following manner.
For two objects
, a morphism
, and the images
, and
,
let (non-strict) precedence be defined by
(5.9)
and let equality be defined by
(5.10)
if the induced epimorphism
is an isomorphism.
Corollary 5.1. If both components of the pairs
are restricted to Hopfian groups, then the pre-order of transfer targets
is actually a partial order.
Proof. We use the functorial properties of the functor F. The reflexivity of the partial order follows from the functorial identity in Formula (7.14), and the transitivity is a consequence of the functorial compositum in Formula (7.15), given in the appendix. The antisymmetry might be a problem for infinite groups, since it is known that there exist so-called non-Hopfian groups. However, for finite groups, and more generally for Hop- fian groups, it is due to the implication
. □
5.2. Singulets of Transfer Kernels
Suppose that G and T are groups,
is the image of G under a homomorphism
, and
is the image of a subgroup
of finite index
. Let
be the Artin transfer from G to
.
Theorem 5.2. If
, then the image
of a left transversal
of U in G is a left transversal of V in H, the index
remains the same and is therefore finite, and the Artin transfer
from H to
exists.
1) The following connections exist between the two Artin transfers: the required condition for the composita of mappings in the commutative diagram in Figure 3,
(5.11)
and, consequently, the inclusion of the kernels,
(5.12)
2) A sufficient (but not necessary) condition for the equality of the kernels is given by
(5.13)
![]()
Figure 3. Epimorphism and Artin transfer.
See Figure 3 for a visualization of this scenario.
Proof. The truth of these statements can be justified in the following way. The first part has been proved in
Proposition 2.1 already: Let
be a left transversal of U in G. Then
is a disjoint union but the union
is not necessarily disjoint. For
, we have
![]()
![]()
![]()
![]()
for some
element![]()
![]()
![]()
![]()
. However, if the condition
is satisfied, then we are able to conclude that
, and thus
.
Let
be the epimorphism obtained in the manner indicated in the proof of Theorem 5.1 and Formula (5.2). For the image of
under the Artin transfer, we obtain
![]()
Since
, the right hand side equals
, provided that
is a left transversal of V in H, which is correct when
. This shows that the diagram in Figure 3 is commutative, that is,
. It also yields the connection between the permutations
and the monomials
for all
. As a consequence, we obtain the inclusion
, if
. Finally, if
, then the previous section has shown that
is an isomorphism. Using the inverse isomorphism, we get
, which proves the equation
. More explicitly, we have the following chain of equivalences and implications:
![]()
Conversely,
only implies
. Therefore, we cer-
tainly have
if
, which is, however, not necessary. □
P 5.3. Artin transfers as natural transformations. Artin transfers
can be viewed as components of a natural transformation T between two functors
and F from the following category
to the usual category
of groups.
The objects of the category
are pairs
consisting of a group G and a subgroup
of finite index
,
(5.14)
For two objects
, the set of morphisms
consists of epimor- phisms
satisfying
,
, and the additional condition
for their kernels, briefly written as arrows
,
(5.15)
The forgetful functor
from this category
to the category
of groups maps a pair.
to its first component
, and it maps a morphism
to the underlying epimorphism
.
(5.16)
The functor
from
to the category
of groups maps a pair
to the commutator quotient group
of the subgroup U, and it maps a morphism
to the induced epimorphism
of the restriction
. Note that we must abstain here from letting F map into the subcategory
of abelian groups.
(5.17)
The system T of all Artin transfers fulfils the requirements for a natural transformation
between these two functors, since we have
(5.18)
for every morphism
of the category
.
Definition 5.2. Due to the results in Theorem 5.2, it makes sense to define a pre-order of transfer kernels on the kernels
of the components
of the natural transformation T in the object class
of the category
of groups in the following manner.
For two objects
, a morphism
, and the images
, and
,
let (non-strict) precedence be defined by
(5.19)
and let equality be defined by
(5.20)
if the induced epimorphism
is an isomorphism.
Corollary 5.2. If both components of the pairs
are restricted to Hopfian groups,
then the pre-order of transfer kernels
is actually a partial order.
Proof. Similarly as in the proof of Corollary 5.1, we use the properties of the functor F. The reflexivity is due to the functorial identity in Formula (7.14). The transitivity is due to the functorial compositum in Formula (7.15), where we have to observe the relations
,
, and Formula (7.1) in the appendix for verifying the kernel relation
![]()
additionally to the image relation
![]()
The antisymmetry is certainly satisfied for finite groups, and more generally for Hopfian groups. □
5.3. Multiplets of Transfer Targets and Kernels
Instead of viewing various pairs
which share the same first component G as distinct objects in the categories
, resp.
, which we used for describing singulets of transfer targets, resp. kernels, we now consider a collective accumulation of singulets in multiplets. For this purpose, we shall define a new category
of families, which generalizes the category
, rather than the category
. However, we have to pre- pare this definition with a criterion for the compatibility of a system of subgroups with its image under a homo- morphism.
Proposition 5.1. See also ([7] , Thm.2.3.4, p. 29); ([8] , Satz 3.10, p. 16); ([9] , Thm.2.4, p. 6); ([13] , Thm.X.21, p. 340).
For an epimorphism
of groups, the associated set mappings
(5.21)
are inverse bijections between the following systems of subgroups
(5.22)
Proof. The fourth and fifth statement of Lemma 7.1 in the appendix show that usually the associated set mappings
and
of a homomorphism
are not inverse bijections between systems of sub- groups of G and H. However, if we replace the homomorphism
by an epimorphism with
, then the Formula (7.2) yields the first desired equality
![]()
Guided by the property
of all pre-images
of
, we define a re- stricted system of subgroups of the domain G,
![]()
and, according to Formula (7.1.), we consequently obtain the second required equality
![]()
which yields the crucial pair of inverse set bijections
![]()
□
P 5.4. After this preparation, we are able to specify the new category
. The objects of the category ![]()
are pairs
consisting of a group G and the family of all subgroups
with finite index
,
(5.23)
where I denotes a suitable indexing set. Note that G itself is one of the subgroups
.
The morphisms of the new category are subject to more restrictive conditions, which concern entire families of subgroups instead of just a single subgroup.
For two objects
, the set
of
morphisms consists of epimorphisms
satisfying
, the image conditions
, and the kernel conditions
, which imply the pre-image conditions
, for all
, briefly written as arrows
,
(5.24)
Note that, in view of Proposition 5.1, we can always use the same indexing set I for the domain and for the codomain of morphisms, provided they satisfy the required kernel condition.
Now we come to the essential definition of Artin transfer patterns.
Definition 5.3. Let
be an object of the category
.
The transfer target type (TTT) of G is the family
(5.25)
The transfer kernel type (TKT) of G is the family
(5.26)
The complete Artin pattern of G is the pair
(5.27)
P 5.5. The natural partial order on TTTs and TKTs is reduced to the partial order on the components, according to the Definitions 5.1 and 5.2.
Definition 5.4. Let
be two objects of the category
, where all members of the families
and
are Hopfian groups.
Then (non-strict) precedence of TTTs is defined by
(5.28)
and equality of TTTs is defined by
(5.29)
(Non-strict) precedence of TKTs is defined by
(5.30)
and equality of TKTs is defined by
(5.31)
We partition the indexing set I in two disjoint components, according to whether components of the Artin pattern remain fixed or change under an epimorphism.
Definition 5.5. Let
be two objects of the category
, and let
be a morphism between these objects.
The stable part and the polarized part of the Artin pattern
of G with respect to
are defined by
(5.32)
Accordingly, we have
(5.33)
Note that the precedence of polarized targets is strict as opposed to polarized kernels.
5.4. The Artin Pattern on a Descendant Tree
Firstly, a basic relation
between parent and child (also called immediate descendant), corre- sponding to a directed edge
of the tree, for any vertex
which is different from the root R of the tree.
Secondly, an induced non-strict partial order relation,
for some integer
, between ancestor and descendant, corresponding to a path
of directed edges, for an arbitrary vertex
, that is, the ancestor
is an iterated parent of the descendant. Note that only an empty path with
starts from the root R of the tree, which has no parent.
Just a brief justification of the partial order: Reflexivity is due to the relation
. Transitivity
follows from the rule
. Antisymmetry is a consequence of the absence of cycles, that is,
implies
and thus
.
P 5.7. The category of a tree. Now let
be a rooted directed tree whose vertices are groups
. Then we define
, the category associated with
, as a subcategory of the category
which was introduced in the Formulas (5.23) and (5.24).
The objects of the category
are those pairs
in the object class of the category
whose
first component is a vertex of the tree
,
(5.34)
The morphisms of the category
are selected along the paths of the tree
only.
For two objects
, the set
of morphisms is either empty or consists of a single element only,
(5.35)
In the case of an ancestor-descendant relation between H and G, the specification of the supercategory ![]()
enforces the following constraints on the unique morphism
: the image relations
and the kernel relations
, for all
.
P 5.8. At this position, we must start to be more concrete. In the descendant tree
of a group R, which is the root of the tree, the formal parent operator
gets a second meaning as a natural projection
,
, from the child G onto its parent
, which is always the quotient of G by a suitable normal subgroup
. To be precise, the epimorphism
with kernel
is actually dependent on its domain G. Therefore, the formal power
is only a convenient
abbreviation for the compositum
.
As described in [3] , there are several possible selections of the normal subgroup N in the parent definition
. Here, we would like to emphasize the following three choices of characteristic subgroups N of the child G. If p denotes a prime number and
is the descendant tree of a finite p-group R, then it is usual to take for ![]()
1) either the last non-trivial member
of the lower central series of G
2) or the last non-trivial member
of the lower exponent-p central series of G
3) or the last non-trivial member
of the derived series of G,
where
denotes the nilpotency class,
the lower exponent p-class, and
the derived length of G, respectively.
Note that every descendant tree of finite p-groups is subtree of a descendant tree with abelian root. Therefore, it is no loss of generality to restrict our attention to descendant trees with abelian roots.
Theorem 5.3. A uniform warranty for the comparability of the Artin patterns
of all vertices G of a descendant tree
of finite p-groups with abelian root R, in the sense of the natural partial
order, is given by the following restriction of the family of subgroups
in the corresponding object
of the category
. The restriction depends on the definition of a parent
in the
descendant tree.
1)
for all
, when
with
.
2)
for all
, when
with
.
3)
for all
, when
with
.
Proof. If parents are defined by
with
, then we have
and
for any
. The largest of these kernels arises for
. Therefore, uniform comparability of Artin patterns is warranted by the restriction
for all
.
The parent definition
with
implies
and
for any
. The largest of these kernels arises for
. Consequently, a uniform comparability of Artin patterns is guaranteed by the restriction
for all
.
Finally, in the case of the parent definition
with
, we have
and
for any
. The largest of these kernels arises for
. Consequently, a uniform comparability of Artin patterns is guaranteed by the condition
for all
.
□
P 5.9. Note that the first and third condition coincide since both,
and
, denote the commutator subgroup
. So the family
is restricted to the normal subgroups which contain
, as announced in the paragraph preceding Lemma 3.1.
The second condition restricts the family
to the maximal subgroups of G inclusively the group G
itself and the Frattini subgroup
.
P 5.10. Since we shall mainly be concerned with the first and third parent definition for descendant trees, that is, either with respect to the lower central series or to the derived series, the comparability condition in Theorem 5.3 suggests the definition of a category
whose objects are subject to more severe conditions than those in Formula (5.23),
(5.36)
but whose morphism are defined exactly as in Formula (5.24). The new viewpoint leads to a corresponding modification of Artin transfer patterns.
Definition 5.6. Let
be an object of the category
.
The Artin pattern, more precisely the restricted Artin pattern, of G is the pair
(5.37)
whose components, the TTT and the TKT of G, are defined as in the Formulas (5.25) and (5.26), but now with respect to the smaller system of subgroups of G.
P 5.11. The following Main Theorem shows that any non-metabelian group G with derived length
and finite abelianization
shares its Artin transfer pattern
, in the restricted sense, with its metabelianization, that is the second derived quotient
.
Theorem 5.4. (Main Theorem.) Let G be a (non-metabelian) group with finite abelianization
, and denote by
,
, the terms of the derived series of G, that is
and
for
, in particular,
and
, then
1) every subgroup
which contains the commutator subgroup
is a normal subgroup
of finite index
,
2) for each
, there is a chain of normal subgroups
(5.38)
3) for each
, the targets of the transfers
and
are equal in the sense of the natural order,
(5.39)
4) for each
, the kernels of the transfers
and
are equal in the sense of the natural order,
(5.40)
Proof. We use the natural epimorphism
,
.
1) If U is an intermediate group
, then
is a normal subgroup of G, according to Lemma 3.1. The assumption
implies that
is a divisor of the integer
. Therefore, the Artin transfer
exists.
2) Firstly,
implies
. Since
is characteristic in U, we also have
. Similarly,
is characteristic in
and thus normal in
. Finally, we obtain
.
3) The mapping
,
, is an epimorphism with kernel
. Consequently, the isomorphism theorem in Remark 7.3 of the appendix yields the isomorphism
.
4) Firstly, the restriction
is an epimorphism which induces an isomorphism
, since
and
, according to Theorem 5.1. Secondly, according
to Theorem 5.2, the condition
implies that the index
is finite, the Artin transfer
exists, the composite mappings
commute, and, since we even have
, the transfer kernels satisfy the relation
. In the sense of the natural partial order on transfer kernels this
means equality
, since
and thus, similarly as in Proposition 5.1, the map
establishes a set bijection between the systems of subgroups
and
, where
.
□
Remark 5.1. At this point it is adequate to emphasize how similar concepts in previous publications are related to the concept of Artin patterns. The restricted Artin pattern
in Definition 5.6 was essentially introduced in ([19] , Dfn.1.1, p. 403), for a special case already earlier in ([20] , §1, p. 417). The name Artin pattern appears in ([21] , Dfn.3.1, p. 747) for the first time. The complete Artin pattern
in Definition 5.3 is new in the present article, but we should point out that it includes the iterated IPADs (index-p abelianization data) in ([18] , Dfn.3.5, p. 289) and the iterated IPODs (index-p obstruction data) in ([22] , Dfn.4.5).
In a second remark, we emphasize the importance of the preceding Main Theorem for arithmetical applications.
Remark 5.2. In algebraic number theory, Theorem 5.4 has striking consequences for the determination of the length
of the p-class tower
, that is the maximal unramified pro-p extension, of an algebraic number field K with respect to a given prime number p. It shows the impossibility of deciding, exclusively with the aid of the restricted Artin pattern
, which of several assigned candidates G with distinct derived lengths
is the actual p-class tower group
. (In contrast,
can always be recognized with
.)
This is the point where the complete Artin pattern
enters the stage. Most recent investigations by means of iterated IPADs of 2nd order, whose components are contained in
, enabled decisions between
in [18] [22] .
Another successful method is to employ cohomological results by I.R. Shafarevich on the relation rank
for selecting among several candidates G for the p-class tower group, in dependence on the torsion-free unit rank of the base field K, for instance in [21] [23] .
Important examples for the concepts in §5 are provided in the following subsections.
5.5. Abelianization of Type (p,p)
Let G be a p-group with abelianization
of elementary abelian type
. Then G has
maximal subgroups
of index
. For each
, let
be the Artin transfer homomorphism from G to the abelianization of
.
Definition 5.7. The family of normal subgroups
is called the transfer kernel type
(TKT) of G with respect to
.
Remark 5.3. For brevity, the TKT is identified with the multiplet
, whose integer components
are given by
(5.41)
Here, we take into consideration that each transfer kernel
must contain the commutator subgroup
of G, since the transfer target
is abelian. However, the minimal case
cannot occur, according to Hilbert’s Theorem 94.
A renumeration of the maximal subgroups
and of the transfers
by means of a per-
mutation
gives rise to a new TKT
with respect to
, identified with
, where
![]()
It is adequate to view the TKTs
as equivalent. Since we have
,
the relation between
and
is given by
. Therefore,
is another representative of the orbit
of
under the operation
of the symmetric group
on the set of all mappings from
to
, where the extension
of the permutation
is defined by
, and we formally put
,
.
Definition 5.8. The orbit
of any representative
is an invariant of the p-group G and is called its transfer kernel type, briefly TKT.
Remark 5.4. This definition of
goes back to the origins of the capitulation theory and was introduced by Scholz and Taussky for
in 1934 [24] . Several other authors used this original definition and investigated capitulation problems further. In historical order, Chang in 1977 [25] , Chang and Foote in 1980 [26] , Heider and Schmithals in 1982 [27] , Brink in 1984 [28] , Brink and Gold in 1987 [29] , Nebelung in 1989 [30] , and ourselves in 1991 [31] and in 2012 [32] .
In the brief form of the TKT
, the natural order is expressed by
for
.
Let
denote the counter of total transfer kernels
, which is
an invariant of the group G. In 1980, Chang and Foote [26] proved that, for any odd prime p and for any integer
, there exist metabelian p-groups G having abelianization
of type
such that
. However, for
, there do not exist non-abelian 2-groups G with
, such that
. Such groups must be metabelian of maximal class. Only the elementary abelian 2-group
has
.
In the following concrete examples for the counters
, and also in the remainder of this article, we use identifiers of finite p-groups in the SmallGroups Library by Besche, Eick and O’Brien [33] [34] .
Example 5.1. For
, we have the following TKTs
:
・
for the extra special group
of exponent 9 with
,
・
for the two groups
with
,
・
for the group
with
,
・
for the group
with
,
・
for the extra special group
of exponent 3 with
.
5.6. Abelianization of Type (p2, p)
Let G be a p-group with abelianization
of non-elementary abelian type
. Then G possesses
maximal subgroups
of index
, and
subgroups
of index
. See Figure 1.
P 5.12. Convention. Suppose that
is the distinguished maximal subgroup which is the product of all subgroups of index
, and
is the distinguished subgroup of index ![]()
which is the intersection of all maximal subgroups, that is the Frattini subgroup
of G.
P 5.13. First layer. For each
, let
be the Artin transfer homomorphism from G to the ab- elianization of
.
Definition 5.9. The family
is called the first layer transfer kernel type of G with respect to
and
, and is identified with
, where
(5.42)
Remark 5.5. Here, we observe that each first layer transfer kernel is of exponent p with respect to
and consequently cannot coincide with
for any
, since
is cyclic of order
, whereas
is bicyclic of type
.
P 5.14. Second layer. For each
, let
be the Artin transfer homomorphism from G to the abelianization of
.
Definition 5.10. The family
is called the second layer transfer kernel type of G with respect to
and
, and is identified with
, where
(5.43)
P 5.15. Transfer kernel type.
Combining the information on the two layers, we obtain the (complete) transfer kernel type
![]()
of the p-group G with respect to
and
.
Remark 5.6. The distinguished subgroups
and
are unique invariants of G and should not be renumerated. However, independent renumerations of the remaining maximal subgroups
and the transfers
by means of a permutation
, and of the remaining subgroups
of index
and the transfers
by means of a permutation
, give rise to new TKTs
with respect to
and
, identified with
, where
![]()
and
with respect to
and
, identified with
,
where
![]()
It is adequate to view the TKTs
and
as equivalent. Since we have
![]()
resp.
![]()
the relations between
and
, resp.
and
, are given by
, resp.
. Therefore,
is another representative of the orbit
of
under the operation
![]()
of the product of two symmetric groups
on the set of all pairs of mappings from
to
, where the extensions
and
of a permutation
are defined by
and
, and formally
,
,
, and
.
Definition 5.11. The orbit
of any representative
is an invariant of the p- group G and is called its transfer kernel type, briefly TKT.
P 5.16. Connections between layers.
The Artin transfer
from G to a subgroup
of index
![]()
is the compositum
of the induced transfer
from
to
(in the sense of Corollary 7.1 or Corollary 7.3 in the appendix) and the Artin transfer
from G to
, for any intermediate subgroup
of index
(
). There occur two situations:
・ For the subgroups
only the distinguished maximal subgroup
is an intermediate subgroup.
・ For the Frattini subgroup
all maximal subgroups
are intermediate subgroups.
This causes restrictions for the transfer kernel type
of the second layer,
since
, and thus
・
, for all
,
・ but even
.
Furthermore, when
with
and
, an element
(
) which is of order
with respect to
, can belong to the transfer kernel
only if its pth power
is contained in
, for all intermediate subgroups
, and thus:
・
, for certain
, enforces the first layer TKT singulet
,
・ but
, for some
, even specifies the complete first layer TKT multiplet
, that is
, for all
.
6. Stabilization and Polarization in Descendant Trees
P 6.1. Theorem 5.4 has proved that it suffices to get an overview of the restricted Artin patterns of metabelian groups G with
, since groups G of derived length
will certainly reveal exactly the same patterns as their metabelianizations
.
In this section, we present the complete theory of stabilization and polarization of the restricted Artin patterns for an extensive exemplary case, namely for all metabelian 3-groups G with abelianization
of type (3,3).
Since the bottom layer, resp. the top layer, of the restricted Artin pattern will be considered in Theorem 6.4 on the commutator subgroup
, resp. Theorem 6.5 on the entire group G, we first focus on the intermediate layer
of the maximal subgroups
.
6.1. 3-Groups of Non-Maximal Class
P 6.2. We begin with groups G of non-maximal class. Denoting by m the index of nilpotency of G, we let
with
be the centralizers of two-step factor groups
of the lower central series, that is, the biggest subgroups of G with the property
. They form an as- cending chain of characteristic subgroups of G,
, which contain the commutator subgroup.
coincides with G if and only if
. We characterize the smallest two-step centralizer different from the commutator group by an isomorphism invariant
. According to Nebelung [30] , we can assume that G has order
, class
, and coclass
, where
. Let generators of
be selected such that
,
, if
, and
. Suppose that a fixed ordering of the four maximal subgroups of G is defined by
with
,
,
, and
. Let the main commutator of G be declared by
and higher commutators recursively by
,
for
. Starting with the powers
,
, let
,
for
, and put
,
.
Theorem 6.1. (Non-maximal class.) Let G be a metabelian 3-group of nilpotency class
and coclass
with abelianization
. With respect to the projection
onto the parent, the restricted Artin pattern
of G reveals
1) a bipolarization and partial stabilization, if G is an interface group with bicyclic last lower central equal to the bicyclic first upper central, more precisely
(6.1)
2) a unipolarization and partial stabilization, if G is a core group with cyclic last lower central and bicyclic first upper central, more precisely
(6.2)
3) a nilpolarization and total stabilization, if G is a core group with cyclic last lower central equal to the cyclic first upper central, more precisely
(6.3)
Proof. Theorems 5.1 and 5.2 tell us that for detecting whether stabilization occurs from parent
to child G, we have to compare the projection kernel
with the commutator subgroups
of the four maximal normal subgroups
,
. According to ([35] , Cor.3.2, p. 480) these derived subgroups are given by
(6.4)
provided the generators of G are selected as indicated above. On the other hand, the projection kernel
is given by
(6.5)
Combining this information with
, we obtain the following results.
・
for
if
, independently of
.
・
for
if
, which implies
.
・
but
if
,
, which also implies
.
・
for
if
,
, which implies
.
Taken together, these results justify all claims. □
Example 6.1. Generally, the parent
of an interface group G ([19] , Dfn.3.3, p. 430) with bicyclic last non-trivial lower central is a vertex of a different coclass graph with lower coclass. In the case of a bipolarization ([19] , Dfn.3.2, p. 430), which is now also characterized via the Artin pattern by Formula (6.1) for
, we can express the membership in coclass graphs by the implication: If
with
, then
. A typical example is the group
of coclass 3 with parent
of coclass 2 (again with identifiers in the SmallGroups database [33] [34] ), where
![]()
and
![]()
In contrast, a core group G ([19] , Dfn.3.3, p. 430) with cyclic last non-trivial lower central and its parent
are vertices of the same coclass graph. In dependence on the p-rank of its centre
, the Artin pattern either shows a unipolarization as in Formula (6.2), if the centre is bicyclic, or a total stabilization as in Formula (6.3), if the centre is cyclic. Typical examples are the group
with parent
, both of coclass 2, where the Artin pattern shows a unipolarization
![]()
and
![]()
and the group
with parent
, both of coclass 2, where the Artin pattern shows a total stabilization
![]()
and
![]()
6.2. p-Groups of Maximal Class
P 6.3. Next we consider p-groups of maximal class, that is, of coclass
, but now for an arbitrary prime number
. According to Blackburn [17] and Miech [36] , we can assume that G is a metabelian p-group of order
and nilpotency class
, where
. Then G is of coclass
and the commutator factor group
of G is of type
. The lower central series of G is defined recursively by
and
for
, in particular
.
The centralizer
of the two-step factor group
, that is,
![]()
is the biggest subgroup of G such that
. It is characteristic, contains the commutator subgroup
, and coincides with G, if and only if
. Let the isomorphism invariant
of G be defined by
![]()
where
for
,
for
, and
for
, according to Miech ([36] , p. 331).
Suppose that generators of
are selected such that
, if
, and
.
We define the main commutator
and the higher commutators
for
.
The maximal subgroups
of G contain the commutator subgroup
of G as a normal subgroup of index p and thus are of the shape
. We define a fixed ordering by
and
for
.
Theorem 6.2. (Maximal class.) Let G be a metabelian p-group of nilpotency class
and coclass
, which automatically implies an abelianization
of type
. With respect to the projection
onto the parent, the restricted Artin pattern
of G reveals
1) a unipolarization and partial stabilization, if the first maximal subgroup
of G is abelian, more precisely
(6.6)
2) a nilpolarization and total stabilization, if all four maximal subgroups
of G are non-abelian, more precisely
(6.7)
In both cases, the commutator subgroups of the other maximal normal subgroups of G are given by
(6.8)
Proof. We proceed in the same way as in the proof of Theorem 6.1 and compare the projection kernel
with the commutator subgroups
of the
maximal normal subgroups
,
. According to ([35] , Cor.3.1, p. 476) they are given by
(6.9)
if the generators of G are chosen as indicated previously. The cyclic projection kernel is given uniformly by
(6.10)
Using the relation
, we obtain the following results.
・
for
if
.
・
if and only if
, that is,
.
The claims follow by applying Theorems 5.1 and 5.2. □
Example 6.2. For
, typical examples are the group
with parent
, both of coclass 1, where the Artin pattern shows a unipolarization ([19] , Dfn.3.1, p. 413)
![]()
and
![]()
and the group
with parent
, both of coclass 1, where the Artin pattern shows a total stabilization
![]()
and
![]()
6.3. Extreme Interfaces of p-Groups
P 6.4. Finally, what can be said about the extreme cases (excluded in Theorems 6.1 and 6.2) of non-abelian p-groups having the smallest possible nilpotency class
for coclass
and
for coclass
? In these particular situations, the answers can be given for arbitrary prime numbers
.
Theorem 6.3. Let G be a metabelian p-group with abelianization
of type
.
1) If G is of coclass
and nilpotency class
, then
must be odd and the coclass must be
exactly.
2) If G is of coclass
and nilpotency class
, then G is an extra special p-group of order
and exponent p or
.
In both cases, there occurs a total polarization and no stabilization at all, more explicitly
(6.11)
Proof. Suppose that G is a metabelian p-group with
.
1) According to O. Taussky [37] , a 2-group G with abelianization
of type
must be of coclass
. Consequently,
implies
.
Since the minimal nilpotency class c of a non-abelian group with coclass
is given by
, the case
cannot occur for
.
So we are considering metabelian p-groups G with
, nilpotency class
and coclass
for odd
, which form the stem of the isoclinism family
in the sense of P. Hall. According to ([19] , Lem.3.1, p. 446), the commutator subgroups
of the maximal subgroups
are cyclic of degree p, for such a group
. However, the kernel of the parent projection
is the bicyclic group
of type
([19] , §3.5, p. 445), which cannot be contained in any of the cyclic
with
.
2) According to ([35] , Cor.3.1, p. 476), the commutator subgroups
of all maximal subgroups
are trivial, for a metabelian p-group G of coclass
and nilpotency class
, which implies
. Thus, the kernel of the parent projection
is not contained in any
.
In both cases, the final claim is a consequence of the Theorems 5.1 and 5.2. □
Example 6.3. For
, a typical example for the interface between groups of coclass 2 and 1 is the group
of coclass 2 with parent
of coclass 1, where the Artin pattern shows a total polarization
![]()
and
![]()
For
, a typical example for the interface between non-abelian and abelian groups is the extra special quaternion group
with parent
both of coclass 1, where the Artin pattern shows a total polarization
![]()
and
![]()
Summarizing, we can say that the last three Theorems 6.1, 6.2, and 6.3 underpin the fact that Artin transfer patterns provide a marvellous tool for classifying finite p-groups.
6.4. Bottom and Top Layer of the Artin Pattern
P 6.5. We conclude this section with supplementary general results concerning the bottom layer and top layer of the restricted Artin pattern.
Theorem 6.4. (Bottom layer.) The type of the commutator subgroup
can never remain stable for a metabelian vertex
of a descendant tree
with respect to the lower central series, lower exponent-p central series, or derived series. The kernel of
is equal to G (Principal Ideal Theorem).
Proof. All possible kernels
, resp.
, resp.
, of the parent projections
are non trivial, and can therefore never be contained in the trivial second derived subgroup G". According to Theorem 5.1, the type of the commutator subgroup G' cannot be stable. The Principal Ideal Theorem is due to Furtwängler [38] and is also proved in ([13] Thm.10.18, p. 313). □
Example 6.4. In Example 6.1, we point out that the group
with cyclic centre and its parent
, both of coclass 2, cannot be distinguished by their TTT
![]()
and TKT
![]()
due to a total stabilization of the restricted Artin pattern as in Formula (6.3). However, the type of their commutator subgroup (the second layer of their TTT) admits a distinction, since
![]()
Theorem 6.5. (Top layer.) In a descendant tree
with respect to the lower central series or derived series, the type of the abelianization
of
remains stable. The kernel of
is equal to
.
Proof. This follows from Theorem 5.1, since even the maximal possible kernel
, resp.
, of the parent projections
is contained in the commutator subgroup
of G.
□
We briefly emphasize the different behaviour of trees where parents are defined with the lower exponent-p central series.
Theorem 6.6. In a descendant tree
with respect to the lower exponent-p central series, only the p-rank of the abelianization
of the vertices
remains stable.
Proof. Denote by
the p-rank of the abelianization of G. According to Theorem 5.1, the maximal possible kernel
of the parent projections
is the Frattini subgroup which is contained in all maximal subgroups
of G. According to Proposition 5.1, the map
induces a bijection between the sets of maximal subgroups of the child G and the parent
, whose cardinality is given by
. Consequently, we have
. □
Acknowledgements
The author would like to express his heartfelt gratitude to Professor Mike F. Newman from the Australian National University in Canberra, Australian Capital Territory, for his continuing encouragement and interest in our endeavour to strengthen the bridge between group theory and class field theory which was initiated by the ideas of Emil Artin, and for his untiring willingness to share his extensive knowledge and expertise and to be a source of advice in difficult situations.
We also gratefully acknowledge that our research is supported by the Austrian Science Fund (FWF): P 26008- N25.
Appendix: Induced Homomorphism between Quotient Groups
Throughout this appendix, let
be a homomorphism from a source group (domain) G to a target group (codomain) H.
A.1. Image, Pre-Image and Kernel
P 7.1. First, we recall some basic facts concerning the image and pre-image of normal subgroups and the kernel of the homomorphism
.
Lemma 7.1. Suppose that
and
are subgroups, and
are elements.
1) If
is a normal subgroup of G, then its image
is a normal subgroup of the (total) image
.
2) If
is a normal subgroup of the image
, then the pre-image
is a normal subgroup of G.
In particular, the kernel
of
is a normal subgroup of G.
3) If
, then there exists an element
such that
.
4) If
, then
, i.e., the pre-image of the image satisfies
(7.1)
5) Conversely, the image of the pre-image is given by
(7.2)
The situation of Lemma 7.1 is visualized by Figure 4, where we briefly write
and
.
Remark 7.1. Note that, in the first statement of Lemma 7.1, we cannot conclude that
is a normal subgroup of the target group H, and in the second statement of Lemma 7.1, we need not require that
is a normal subgroup of the target group H.
Proof. 1) If
, then
for all
,
and thus
for all
, i.e.,
.
2) If
, then
, that is,
. In particular, we have
, i.e.,
, and
consequently
.
To prove the claim for the kernel, we put
.
3) If
, then
, and thus
. (See also [7] , Thm.2.2.1, p. 27).
4) If
, then
, and thus
, by (3). This shows
, and the opposite inclusion is obvious.
![]()
Figure 4. Kernel, image and pre-image under a homomorphism f.
Finally, since
is normal, we have
.
5) This is a consequence of the properties of the set mappings
and
associated with the homomorphism
.
□
A.2. Criteria for the Existence of the Induced Homomorphism
P 7.2. Now we state the central theorem which provides the foundation for lots of useful applications. It is the most general version of a series of related theorems, which is presented in Bourbaki ([14] , Chap.1), Structures algëriques, Prop.5, p. A I.35]. Weaker versions will be given in the subsequent corollaries.
Theorem 7.1. (Main Theorem)
Suppose that
is a normal subgroup of G and
is a normal subgroup of H. Let
and
denote the canonical projections onto the quotients.
・ The following three conditions for the homomorphism
are equivalent.
1) There exists an induced homomorphism
such that
, that is,
(7.3)
2)
.
3)
.
・ If the induced homomorphism
of the quotients exists, then it is determined uniquely by
, and its kernel, image and cokernel are given by
(7.4)
In particular,
is a monomorphism if and only if
.
Moreover,
is an epimorphism if and only if
.
In particular,
is certainly an epimorphism if
is onto.
We summarize the criteria for the existence of the induced homomorphism in a formula:
(7.5)
The situation of Theorem 7.1 is shown in the commutative diagram of Figure 5.
Remark 7.2. If the normal subgroup
in the assumptions of Theorem 7.17 is taken as
, then the induced homomorphism
exists automatically and is a monomorphism.
![]()
Figure 5. Induced homomorphism
of quotients.
Note that
does not imply
but only
, if
is not an epimorphism. Similarly,
does not imply
but only
, if
is not a monomorphism.
Proof.
・ (1) Þ (2): If there exists a homomorphism
such that
for all
, then, for any
, we have
, and thus
, which means
. It follows that
.
(2) Þ (1): If
, then the image
of the coset
under
is independent of the re- presentative
: If
for
, then
and thus
. Consequently, we have
. Furthermore,
is a homomorphism, since
![]()
(2) Þ (3): If
, then
.
(3) Þ (2): If
, then
.
・ The image of any
under
is determined uniquely by
, since
.
The kernel of
is given by
, and for
we have
![]()
that is
, which clearly contains
, since
.
The cokernel of
is given by
, if
.
Finally, if
is an epimorphism, then
is also an epimorphism, which forces the terminal map
to be an epimorphism. □
A.3. Factorization through a Quotient
P 7.3. Theorem 7.1 can be used to derive numerous special cases. Usually it suffices to consider the quotient group
corresponding to a normal subgroup U of the source group G of the homomorphism
and to view the target group H as the trivial quotient H/1. In this weaker form, the existence criterion for the induced homomorphism occurs in Lang’s book ([39] , p. 17).
Corollary 7.1. (Factorization through a quotient)
Suppose
is a normal subgroup of G and
denotes the natural epimorphism onto the quotient.
If
, then there exists a unique homomorphism
such that
, that is,
for all
.
Moreover, the kernel of
is given by
.
Again we summarize the criterion in a formula:
(7.6)
In this situation the homomorphism
is said to factor or factorize through the quotient
via the canonical projection
and the induced homomorphism
.
The scenario of Corollary 7.1 is visualized by Figure 6.
Proof. The claim is a consequence of Theorem 7.1 in the special case that
is the trivial group. The equivalent conditions for the existence of the induced homomorphism
are
resp.
. □
Remark 7.3. Note that the well-known isomorphism theorem (sometimes also called homomorphism theorem) is a special case of Corollary 7.1. If we put
and if we assume that
is an epimorphism with
, then the induced homomorphism
is an isomorphism, since
.
In this weakest form,
(7.7)
![]()
Figure 6. Homomorphism f factorized through a quotient.
actually without any additional assumptions being required, the existence theorem for the induced homomorphism appears in almost every standard text book on group theory or algebra, e.g., ([7] , Thm.2.3.2, p. 28) and ([13] , Thm.X.18, p. 339).
A.4. Application to Series of Characteristic Subgroups
P 7.4. The normal subgroup
in the assumptions of Corollary 7.1 can be specialized to various characteristic subgroups of G for which the condition
can be expressed differently, namely by invariants of series of characteristic subgroups.
Corollary 7.2. The homomorphism
can be factorized through various quotients of G in the following way. Let n be a positive integer and p be a prime number.
1)
factors through the nth derived quotient
if and only if the derived length of
is bounded by
.
2)
factors through the nth lower central quotient
if and only if the nilpotency class of
is bounded by
.
3)
factors through the nth lower exponent-p central quotient
if and only if the p-class of
is bounded by
.
We summarize these criteria in terms of the length of series in a formula:
(7.8)
Proof. By induction, we show that, firstly,
,
secondly,
,
and finally, ![]()
.
Now, the claims follow from Corollary 7.1 by observing that
iff
,
iff
, and
iff
□
The following special case is particularly well known. Here we take the commutator subgroup
of G as our charecteristic subgroup, which can either be viewed as the term
of the lower central series of G or as the term
of the derived series of G.
Corollary 7.3. A homomorphism
passes through the derived quotient
of its source group G if and only if its image
is abelian.
Proof. Putting
in the first statement or
in the second statement of Corollary 7.2 we obtain the well-known special case that
passes through the abelianization
if and only if
is abelian, which is equivalent to
, and also to
. □
The situation of Corollary 7.3 is visualized in Figure 7.
Using the first part of the proof of Corollary 7.2 we can recognize the behavior of several central series under homomorphisms.
Lemma 7.2. Let
be a homomorphism of groups and suppose that
is an integer and
a prime number. Let
be a subgroup with image
.
1) If
, then
![]()
2) If
, then
![]()
3) If
, then
![]()
Proof. 1) Let
, then
and
. Consequently, we have
if
and
if
.
2) Let
, then
and
. Thus, we have
if
and
if
.
3) Let
, then
and
. Therefore, we have
if
and
if
.□
A.5. Application to Automorphisms
Corollary 7.4. (Induced automorphism)
![]()
Figure 7. Homomorphism
passing through the derived quotient.
Let
be an epimorphism of groups,
, and assume that
is an auto- morphism of G.
1) There exists an induced epimorphism
such that
, if and only if
, resp.
.
2) The induced epimorphism
is also an automorphism of H,
, if and only if
(7.9)
In the second statement,
is said to have the kernel invariance property (KIP) with respect to
.
The situation of Corollary 7.4 is visualized in Figure 8.
Proof. Since
is supposed to be an epimorphism, the well-known isomorphism theorem in Remark 7.3 yields a representation of the image
as a quotient.
1) According to Theorem 7.1, the automorphism
, simply viewed as a homomorphism
, induces a homomorphism
if and only if
. Since
is an epimorphism,
is also an epimorphism with kernel
.
2) Finally,![]()
![]()
![]()
![]()
.
□
Remark 7.4. If
is a characteristic subgroup of G, then Corollary 7.4 makes sure that any automorphism
induces an automorphism
, where
. The reason is that, by definition, a characteristic subgroup of G is invariant under any automorphism of G.
P 7.5. We conclude this section with a statement about GI-automorphisms (generator-inverting auto- morphisms) which have been introduced by Boston, Bush and Hajir ([40] , Dfn.2.1). The proof requires results of Theorem 7.1, Corollary 7.4, and Corollary 7.2.
Theorem 7.2. (Induced generator-inverting automorphism)
Let
be an epimorphism of groups with
, and assume that
is an automorphism satisfying the KIP
, and thus induces an automorphism
.
If
is generator-inverting, that is,
(7.10)
then
is also generator-inverting, that is,
for all
.
Proof. According to Corollary 7.4,
induces an automorphism
, since
.
Two applications of the Remark 7.4 after Corollary 7.4, yield:
induces an automorphism
, since
is characteristic in G, and
induces an automorphism
, since
is characteristic in H.
Using Theorem 7.1 and the first part of the proof of Corollary 7.2, we obtain:
induces an epimorphism
, since
.
The actions of the various induced homomorphisms are given by
for
,
for
,
for
, and
for
.
Finally, combining all these formulas and expressing
for a suitable
, we see that
implies the required relation for
:
□
A.6. Functorial Properties
P 7.6. The mapping
which maps a homomorphism of one category to an induced homomorphism of another category can be viewed as a functor F.
In the special case of induced homomorphisms
between quotient groups, we define the domain of the functor F as the following category
.
The objects of the category are pairs
consisting of a group G and a normal subgroup
,
(7.11)
For two objects
, the set of morphisms
consists of homomorphisms
such that
, briefly written as arrows
,
(7.12)
The functor
from this new category
to the usual category
of groups
maps a pair
to the corresponding quotient group
,and it maps a morphism
to the induced homomorphism
,
(7.13)
Existence and uniqueness of
have been proved in Theorem 7.1 under the assumption that
, which is satisfied according to the definition of the arrow
.
The functorial properties, which are visualized in Figure 9, can be expressed in the following form.
![]()
Figure 9. Functorial properties of induced homomorphisms.
Firstly, F maps the identity morphism
having the trivial property
to the identity homomorphism
(7.14)
and secondly, F maps the compositum
of two morphisms
and
, which obviously enjoys the required property
![]()
to the compositum
(7.15)
of the induced homomorphisms in the same order.
The last fact shows that F is a covariant functor.
NOTES
![]()
*Respectfully dedicated to Professor M. F. Newman.