_{1}

^{*}

Based on a thorough theory of the Artin transfer homomorphism
from a group
*G* to the abelianization
of a subgroup
of finite index
, and its connection with the permutation representation
and the monomial representation
of
*G*, the Artin pattern
, which consists of families
, resp.
, of transfer targets, resp. transfer kernels, is defined for the vertices
of any descendant tree T of finite
*p*-groups. It is endowed with partial order relations
and
, which are compatible with the parent-descendant relation
of the edges
of the tree T. The partial order enables termination criteria for the
*p*-group generation algorithm which can be used for searching and identifying a finite
*p*-group
*G*, whose Artin pattern
is known completely or at least partially, by constructing the descendant tree with the abelianization
of
*G* as its root. An appendix summarizes details concerning induced homomorphisms between quotient groups, which play a crucial role in establishing the natural partial order on Artin patterns
and explaining the stabilization, resp. polarization, of their components in descendant trees T of finite
*p*-groups.

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 ([

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 ([

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 [

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 [

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 [

Let G be a group and

Definition 2.1. See also ([

1). A left transversal of H in G is an ordered system

G such that

2). Similarly, a right transversal of H in G is an ordered system

cosets of H in G such that

Remark 2.1. For any transversal of H in G, there exists a unique subscript

Lemma 2.1. See also ([

1). If G is non-abelian and H is not a normal subgroup of G, then we can only say that the inverse elements

2). However, if

Proof. 1). Since the mapping

2). For a normal subgroup

Let

Proposition 2.1. The following two conditions are equivalent.

1).

index

2).

We emphasize this important equivalence in a formula:

Proof. By assumption, we have the disjoint left coset decomposition

statements simultaneously.

Firstly, the group

and secondly, any two distinct cosets have an empty intersection

Due to the properties of the set mapping associated with

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

Suppose that

then we have

Multiplying by

Since

Conversely, we use contraposition.

If the kernel

But then the homomorphism

to equal cosets

□

P 2.1. Suppose

for each

Similarly, if

for each

The elements

Definition 2.2 See also ([

The mapping

Lemma 2.2. For the special right transversal

Proof. For the right transversal

On the other hand, for the left transversal

This relation simultaneously shows that, for any

for each

Let G be a group and

Definition 3.3. See also ([

The Artin transfer

resp.

for

Remark 3.1. I.M. Isaacs [

the pre-transfer from G to H. The pre-transfer can be composed with a homomorphism

an abelian group A to define a more general version of the transfer

natural epimorphism

Assume that

Proposition 3.1. See also ([

The Artin transfers with respect to (g) and

Proof. There exists a unique permutation

for all

for all

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

Proposition 3.2. The Artin transfers with respect to

Proof. Using (2.4) in Lemma 2.2 and the commutativity of

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. □

Let

Theorem 3.1. See also ([

The Artin transfer

Proof. Let

This relation simultaneously shows that the Artin transfer

Let

Definition 3.2. The mapping

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

The law reminds of the crossed homomorphisms

These peculiar structures can also be interpreted by endowing the cartesian product

Definition 3.3. For

Theorem 3.2. See also ([

This law of composition on

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

The permutation representation cannot be injective if G is infinite or at least of an order bigger than

Remark 3.2. Formula (3.4) is an example for the left-sided variant of the wreath product on

which implies that the permutation representation

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 ([

Whereas B. Huppert ([

Let G be a group with nested subgroups

Theorem 3.3. See also ([

Then the Artin transfer

This can be seen in the following manner.

Proof. If

respect to K. See also ([

Then, using Corollary 7.3, we have

For each pair of subscripts

resp.

□

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

The preceding proof has shown that

on the first component i (via the permutation

The permutations

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

for all

This law reminds of the chain rule

xÎE of the compositum of differentiable functions

The above considerations establish a third representation, the stabilizer representation,

of the group G in the wreath product

Formula (3.7) proves the following statement.

Theorem 3.4. The stabilizer representation

the wreath product

Let

Theorem 3.5. See also ([

If the permutation

cycles

for

Proof. The reason for this fact is that we obtain another left transversal of H in G by putting

is a disjoint decomposition of G into left cosets of H.

Let us fix a value of

However, for

Consequently,

□

P 3.3. The cycle decomposition corresponds to a double coset decomposition

G modulo the cyclic group

P 3.4. Now let

Theorem 3.6. See also ([

Then the image of

Proof.

can be refined to a left transversal

of H in G. Hence, the formula for the image of x under the Artin transfer

with exponent f independent of j. □

Corollary 3.1. See also ([

In particular, the inner transfer of an element

with the trace element

of H in G as symbolic exponent.

The other extreme is the outer transfer of an element

Proof. The inner transfer of an element

with the trace element

of H in G as symbolic exponent.

The outer transfer of an element

□

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

Lemma 3.1. All subgroups

Proof. Let

Explicit implementations of Artin transfers in the simplest situations are presented in the following section.

P 4.1. Let G be a pro-p group with abelianization

subgroup

commutator subgroup

For each

such that

A convenient selection is given by

Then, for each

according to Equation (3.13) of Corollary 3.1, which can also be expressed by a product of two pth powers,

and to implement the outer transfer as a complete pth power by

according to Equation (3.15) of Corollary 3.1. The reason is that

It should be pointed out that the complete specification of the Artin transfers

P 4.2. Let G be a pro-p group with abelianization

For each

We begin by considering the first layer of subgroups. For each of the normal subgroups

These are the cases where the factor group

Further, a generator

Then, for each

which equals

since

Now we continue by considering the second layer of subgroups. For each of the normal subgroups

such that

Since

and for the outer transfer

exceptionally

and

for

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

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.

Theorem 5.1. Let G and T be groups. Suppose that

1) The commutator subgroup of V is the image of the commutator subgroup of U, that is

2) The restriction

Thus, the abelianization of V,

is an epimorphic image of the abelianization of U, namely the quotient of

3) Moreover, the map

See

Proof. The statements can be seen in the following manner. The image of the commutator subgroup is given by

The homomorphism

Thus,

P 5.2. Functor of derived quotients. In analogy to section §7.6 in the appendix, a covariant functor

In the present situation, we denote by

For two objects

The functor

it maps a morphism

Existence and uniqueness of

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

For two objects

let (non-strict) precedence be defined by

and let equality be defined by

if the induced epimorphism

Corollary 5.1. If both components of the pairs

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

Suppose that G and T are groups,

Theorem 5.2. If

1) The following connections exist between the two Artin transfers: the required condition for the composita of mappings in the commutative diagram in

and, consequently, the inclusion of the kernels,

2) A sufficient (but not necessary) condition for the equality of the kernels is given by

See

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

element

Let

Since

Conversely,

tainly have

P 5.3. Artin transfers as natural transformations. Artin transfers

The objects of the category

For two objects

The forgetful functor

The functor

The system T of all Artin transfers fulfils the requirements for a natural transformation

for every morphism

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

For two objects

let (non-strict) precedence be defined by

and let equality be defined by

if the induced epimorphism

Corollary 5.2. If both components of the pairs

then the pre-order of transfer kernels

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

additionally to the image relation

The antisymmetry is certainly satisfied for finite groups, and more generally for Hopfian groups. □

Instead of viewing various pairs

Proposition 5.1. See also ([

For an epimorphism

are inverse bijections between the following systems of subgroups

Proof. The fourth and fifth statement of Lemma 7.1 in the appendix show that usually the associated set mappings

Guided by the property

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

are pairs

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

morphisms consists of epimorphisms

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

The transfer target type (TTT) of G is the family

The transfer kernel type (TKT) of G is the family

The complete Artin pattern of G is the pair

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

Then (non-strict) precedence of TTTs is defined by

and equality of TTTs is defined by

(Non-strict) precedence of TKTs is defined by

and equality of TKTs is defined by

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

The stable part and the polarized part of the Artin pattern

Accordingly, we have

Note that the precedence of polarized targets is strict as opposed to polarized kernels.

P 5.6. Before we specialize to the usual kinds of descendant trees of finite p-groups ([

Firstly, a basic relation

Secondly, an induced non-strict partial order relation,

Just a brief justification of the partial order: Reflexivity is due to the relation

follows from the rule

P 5.7. The category of a tree. Now let

The objects of the category

first component is a vertex of the tree

The morphisms of the category

For two objects

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

P 5.8. At this position, we must start to be more concrete. In the descendant tree

abbreviation for the compositum

As described in [

1) either the last non-trivial member

2) or the last non-trivial member

3) or the last non-trivial member

where

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

order, is given by the following restriction of the family of subgroups

descendant tree.

1)

2)

3)

Proof. If parents are defined by

The parent definition

Finally, in the case of the parent definition

□

P 5.9. Note that the first and third condition coincide since both,

The second condition restricts the family

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

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

The Artin pattern, more precisely the restricted Artin pattern, of G is the pair

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

Theorem 5.4. (Main Theorem.) Let G be a (non-metabelian) group with finite abelianization

1) every subgroup

2) for each

3) for each

4) for each

Proof. We use the natural epimorphism

1) If U is an intermediate group

2) Firstly,

3) The mapping

4) Firstly, the restriction

to Theorem 5.2, the condition

means equality

□

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 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

This is the point where the complete Artin pattern ^{nd} order, whose components are contained in

Another successful method is to employ cohomological results by I.R. Shafarevich on the relation rank

Important examples for the concepts in §5 are provided in the following subsections.

Let G be a p-group with abelianization

Definition 5.7. The family of normal subgroups

(TKT) of G with respect to

Remark 5.3. For brevity, the TKT is identified with the multiplet

are given by

Here, we take into consideration that each transfer kernel

A renumeration of the maximal subgroups

mutation

It is adequate to view the TKTs

the relation between

Definition 5.8. The orbit

Remark 5.4. This definition of

In the brief form of the TKT

Let

an invariant of the group G. In 1980, Chang and Foote [

In the following concrete examples for the counters

Example 5.1. For

・

・

・

・

・

Let G be a p-group with abelianization

P 5.12. Convention. Suppose that

which is the intersection of all maximal subgroups, that is the Frattini subgroup

P 5.13. First layer. For each

Definition 5.9. The family

Remark 5.5. Here, we observe that each first layer transfer kernel is of exponent p with respect to

P 5.14. Second layer. For each

Definition 5.10. The family

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

Remark 5.6. The distinguished subgroups

and

where

It is adequate to view the TKTs

resp.

the relations between

of the product of two symmetric groups

Definition 5.11. The orbit

P 5.16. Connections between layers.

The Artin transfer

is the compositum

・ For the subgroups

・ For the Frattini subgroup

This causes restrictions for the transfer kernel type

since

・

・ but even

Furthermore, when

・

・ but

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

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

Since the bottom layer, resp. the top layer, of the restricted Artin pattern will be considered in Theorem 6.4 on the commutator subgroup

P 6.2. We begin with groups G of non-maximal class. Denoting by m the index of nilpotency of G, we let

Theorem 6.1. (Non-maximal class.) Let G be a metabelian 3-group of nilpotency class

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

2) a unipolarization and partial stabilization, if G is a core group with cyclic last lower central and bicyclic first upper central, more precisely

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

Proof. Theorems 5.1 and 5.2 tell us that for detecting whether stabilization occurs from parent

provided the generators of G are selected as indicated above. On the other hand, the projection kernel

Combining this information with

・

・

・

・

Taken together, these results justify all claims. □

Example 6.1. Generally, the parent

and

In contrast, a core group G ([

and

and the group

and

P 6.3. Next we consider p-groups of maximal class, that is, of coclass

The centralizer

is the biggest subgroup of G such that

where

Suppose that generators of

We define the main commutator

The maximal subgroups

Theorem 6.2. (Maximal class.) Let G be a metabelian p-group of nilpotency class

1) a unipolarization and partial stabilization, if the first maximal subgroup

2) a nilpolarization and total stabilization, if all four maximal subgroups

In both cases, the commutator subgroups of the other maximal normal subgroups of G are given by

Proof. We proceed in the same way as in the proof of Theorem 6.1 and compare the projection kernel

if the generators of G are chosen as indicated previously. The cyclic projection kernel is given uniformly by

Using the relation

・

・

The claims follow by applying Theorems 5.1 and 5.2. □

Example 6.2. For

and

and the group

and

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

Theorem 6.3. Let G be a metabelian p-group with abelianization

1) If G is of coclass

2) If G is of coclass

In both cases, there occurs a total polarization and no stabilization at all, more explicitly

Proof. Suppose that G is a metabelian p-group with

1) According to O. Taussky [

Since the minimal nilpotency class c of a non-abelian group with coclass

So we are considering metabelian p-groups G with

2) According to ([

In both cases, the final claim is a consequence of the Theorems 5.1 and 5.2. □

Example 6.3. For

and

For

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.

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

Proof. All possible kernels

Example 6.4. In Example 6.1, we point out that the group

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

Proof. This follows from Theorem 5.1, since even the maximal possible kernel

□

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

Proof. Denote by

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.

Daniel C.Mayer, (2016) Artin Transfer Patterns on Descendant Trees of Finite p-Groups. Advances in Pure Mathematics,06,66-104. doi: 10.4236/apm.2016.62008

Throughout this appendix, let

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

1) If

2) If

In particular, the kernel

3) If

4) If

5) Conversely, the image of the pre-image is given by

The situation of Lemma 7.1 is visualized by

Remark 7.1. Note that, in the first statement of Lemma 7.1, we cannot conclude that

Proof. 1) If

and thus

2) If

consequently

To prove the claim for the kernel, we put

3) If

4) If

Finally, since

5) This is a consequence of the properties of the set mappings

□

A.2. Criteria for the Existence of the Induced HomomorphismP 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 ([

Theorem 7.1. (Main Theorem)

Suppose that

・ The following three conditions for the homomorphism

1) There exists an induced homomorphism

2)

3)

・ If the induced homomorphism

In particular,

Moreover,

In particular,

We summarize the criteria for the existence of the induced homomorphism in a formula:

The situation of Theorem 7.1 is shown in the commutative diagram of

Remark 7.2. If the normal subgroup

Note that

Proof.

・ (1) Þ (2): If there exists a homomorphism

(2) Þ (1): If

(2) Þ (3): If

(3) Þ (2): If

・ The image of any

The kernel of

that is

The cokernel of

Finally, if

P 7.3. Theorem 7.1 can be used to derive numerous special cases. Usually it suffices to consider the quotient group

Corollary 7.1. (Factorization through a quotient)

Suppose

If

Moreover, the kernel of

Again we summarize the criterion in a formula:

In this situation the homomorphism

The scenario of Corollary 7.1 is visualized by

Proof. The claim is a consequence of Theorem 7.1 in the special case that

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

In this weakest form,

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., ([

P 7.4. The normal subgroup

Corollary 7.2. The homomorphism

1)

2)

3)

We summarize these criteria in terms of the length of series in a formula:

Proof. By induction, we show that, firstly,

secondly,

and finally,

Now, the claims follow from Corollary 7.1 by observing that

The following special case is particularly well known. Here we take the commutator subgroup

Corollary 7.3. A homomorphism

Proof. Putting

The situation of Corollary 7.3 is visualized in

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

1) If

2) If

3) If

Proof. 1) Let

2) Let

3) Let

Corollary 7.4. (Induced automorphism)

Let

1) There exists an induced epimorphism

2) The induced epimorphism

In the second statement,

The situation of Corollary 7.4 is visualized in

Proof. Since

1) According to Theorem 7.1, the automorphism

2) Finally,

□

Remark 7.4. If

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 ([

Theorem 7.2. (Induced generator-inverting automorphism)

Let

If

then

Proof. According to Corollary 7.4,

Two applications of the Remark 7.4 after Corollary 7.4, yield:

Using Theorem 7.1 and the first part of the proof of Corollary 7.2, we obtain:

The actions of the various induced homomorphisms are given by

Finally, combining all these formulas and expressing

P 7.6. The mapping

In the special case of induced homomorphisms

The objects of the category are pairs

For two objects

The functor

maps a pair

Existence and uniqueness of

The functorial properties, which are visualized in

Firstly, F maps the identity morphism

and secondly, F maps the compositum

to the compositum

of the induced homomorphisms in the same order.

The last fact shows that F is a covariant functor.