Artin transfer patterns on descendant trees of finite p-groups

Based on a thorough theory of the Artin transfer homomorphism \(T_{G,H}:\,G\to H/H^\prime\) from a group \(G\) to the abelianization \(H/H^\prime\) of a subgroup \(H\le G\) of finite index \(n=(G:H)\), and its connection with the permutation representation \(G\to S_n\) and the monomial representation \(G\to H\wr S_n\) of \(G\), the Artin pattern \(G\mapsto(\tau(G),\varkappa(G))\), which consists of families \(\tau(G)=(H/H^\prime)_{H\le G}\), resp. \(\varkappa(G)=(\ker(T_{G,H}))_{H\le G}\), of transfer targets, resp. transfer kernels, is defined for the vertices \(G\in\mathcal{T}\) of any descendant tree \(\mathcal{T}\) of finite \(p\)-groups. It is endowed with partial order relations \(\tau(\pi(G))\le\tau(G)\) and \(\varkappa(\pi(G))\ge\varkappa(G)\), which are compatible with the parent-descendant relation \(\pi(G)<G\) of the edges \(G\to\pi(G)\) of the tree \(\mathcal{T}\). 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 \((\tau(G),\varkappa(G))\) is known completely or at least partially, by constructing the descendant tree with the abelianization \(G/G^\prime\) 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 \((\tau(G),\varkappa(G))\) and explaining the stabilization, resp. polarization, of their components in descendant trees \(\mathcal{T}\) of finite \(p\)-groups.


Introduction
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 descendant trees [26, §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 [32] and O'Brien [33].
The organization of this article is as follows. The detailed theory of Artin transfers 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 [36] and Artin [2] and the relevant sections of the text books by Hall [15], Huppert [18], Gorenstein [14], Aschbacher [3], Doerk and Hawkes [13], Smith and Tabachnikova [37], and Isaacs [19].
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, followed by impressive applications in § 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 tool 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 [8], an appendix in § 7 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. (1) A left transversal of H in G is an ordered system (ℓ 1 , . . . , ℓ n ) of representatives for the left cosets of H in G such that G =˙ n i=1 ℓ i H is a disjoint union. (2) Similarly, a right transversal of H in G is an ordered system (r 1 , . . . , r n ) of representatives for the right cosets of H in G such that G =˙ n i=1 Hr i is a disjoint union. Remark 2.1. For any transversal of H in G, there exists a unique subscript 1 ≤ i 0 ≤ n such that ℓ i0 ∈ H, resp. r i0 ∈ H. The element ℓ i0 , resp. r i0 , which represents the principal coset (i.e., the subgroup H itself) may be replaced by the neutral element 1. (1) If G is non-abelian and H is not a normal subgroup of G, then we can only say that the inverse elements (ℓ −1 1 , . . . , ℓ −1 n ) of a left transversal (ℓ 1 , . . . , ℓ n ) form a right transversal of H in G.
(2) However, if H G is a normal subgroup of G, then any left transversal is also a right transversal of H in G.

Proof.
(1) Since the mapping G → G, x → x −1 is an involution, that is a bijection which is its own inverse, we see that For a normal subgroup H G, we have xH = Hx for each x ∈ G.
Let φ : G → T be a group homomorphism and (ℓ 1 , . . . , ℓ n ) be a left transversal of a subgroup H in G with finite index n = (G : H) ≥ 1. We must check whether the image of this transversal under the homomorphism is again a transversal.
Proposition 2.1. The following two conditions are equivalent.
We emphasize this important equivalence in a formula: φ(ℓ i )φ(H) and (φ(G) : φ(H)) = n ⇐⇒ ker(φ) ≤ H Proof. By assumption, we have the disjoint left coset decomposition G =˙ n i=1 ℓ i H which comprises two statements simultaneously. Firstly, the group G = n i=1 ℓ i H is a union of cosets, and secondly, any two distinct cosets have an empty intersection ℓ i H ℓ j H = ∅, for i = j.
Due to the properties of the set mapping associated with φ, the homomorphism φ 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 ker(φ) ≤ H of the homomorphism φ.

2.2.
Permutation representation. Suppose (ℓ 1 , . . . , ℓ n ) is a left transversal of a subgroup H ≤ G of finite index n = (G : H) ≥ 1 in a group G. A fixed element x ∈ G gives rise to a unique permutation λ x ∈ S n of the left cosets of H in G by left multiplication such that (2.2) xℓ i H = ℓ λx(i) H, resp. xℓ i ∈ ℓ λx(i) H, resp. u x (i) := ℓ −1 λx(i) xℓ i ∈ H, for each 1 ≤ i ≤ n.
Similarly, if (r 1 , . . . , r n ) is a right transversal of H in G, then a fixed element x ∈ G gives rise to a unique permutation ρ x ∈ S n of the right cosets of H in G by right multiplication such that (2.3) Hr i x = Hr ρx(i) , resp. r i x ∈ Hr ρx(i) , resp. w x (i) := r i xr −1 ρx(i) ∈ H, for each 1 ≤ i ≤ n.

Artin transfer
Let G be a group and H ≤ G be a subgroup of finite index n = (G : H) ≥ 1. Assume that (ℓ 1 , . . . , ℓ n ), resp. (r 1 , . . . , r n ), is a left, resp. right, transversal of H in G with associated  [34, p.2].) The Artin transfer T G,H : G → H/H ′ from G to the abelianization H/H ′ of H with respect to (ℓ 1 , . . . , ℓ n ), resp. (r 1 , . . . , r n ), is defined by Remark 3.1. I.M. Isaacs [19, p.149] calls the mapping P : , the pre-transfer from G to H. The pre-transfer can be composed with a homomorphism φ : H → A from H into an abelian group A to define a more general version of the G,H . Proof. There exists a unique permutation σ ∈ S n such that ℓ i H = g σ(i) H, for all 1 ≤ i ≤ n. Consequently, h i := ℓ −1 i g σ(i) ∈ H, resp. g σ(i) = ℓ i h i with h i ∈ H, for all 1 ≤ i ≤ n. For a fixed element x ∈ G, there exists a unique permutation γ x ∈ S n such that we have for all 1 ≤ i ≤ n. Therefore, the permutation representation of G with respect to (g 1 , . . . , g n ) is given by γ Finally, due to the commutativity of the quotient group H/H ′ and the fact that σ and λ x 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.
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.
and the permutation representation G → S n , x → λ x are group homomorphisms: T G,H (xy) = T G,H (x) · T G,H (y) and λ xy = λ x • λ y for x, y ∈ G.
Proof. Let x, y ∈ G be two elements with transfer images T G,H (x) = n i=1 ℓ −1 λx(i) xℓ i · H ′ and T G,H (y) = n j=1 ℓ −1 λy(j) yℓ j · H ′ . Since H/H ′ is abelian and λ y is a permutation, we can change the order of the factors in the following product: This relation simultaneously shows that the Artin transfer T G,H and the permutation representation G → S n , x → λ x are homomorphisms, since T G,H (xy) = T G,H (x)·T G,H (y) and λ xy = λ x • λ y , in a covariant way.
It is illuminating to restate the homomorphism property of the Artin transfer in terms of the monomial representation. The images of the factors x, y are given by In the proof of Theorem 3.1, the image of the product xy turned out to be T G,H (xy) = n j=1 ℓ −1 λx(λy (j)) xℓ λy (j) ℓ −1 λy(j) yℓ j · H ′ = n j=1 u x (λ y (j)) · u y (j) · H ′ , which is a very peculiar law of composition discussed in more detail in the sequel.
The law reminds of the crossed homomorphisms x → u x in the first cohomology group H 1 (G, M ) of a G-module M , which have the property u xy = u y x · u y , for x, y ∈ G. These peculiar structures can also be interpreted by endowing the cartesian product H n × S n with a special law of composition known as the wreath product H ≀ S n of the groups H and S n with respect to the set {1, . . . , n}. Definition 3.3. For x, y ∈ G, the wreath product of the associated monomials and permutations is given by This law of composition on H n × S n causes the monomial representation G → H ≀ S n , x → (u x (1), . . . , u x (n); λ x ) 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 H n × S n endowed with the wreath product is given by (1, . . . , 1; 1), where the last 1 means the identity permutation. If (u x (1), . . . , u x (n); λ x ) = (1, . . . , 1; 1), for some x ∈ G, then λ x = 1 and consequently 1 = u Finally, an application of the inverse inner automorphism with g i yields x = 1, as required for injectivity.
The permutation representation cannot be injective if G is infinite or at least of an order bigger than n!, the factorial of n.
Remark 3.2. Formula (3.4) is an example for the left-sided variant of the wreath product on H n × S n . However, we point out that the wreath product with respect to a right transversal (r 1 , . . . , r n ) of H in G appears in its right-sided variant (3.5) (w x (1), . . . , w x (n); ρ x ) · (w y (1), . . . , w y (n); ρ y ) := (w x (1) · w y (ρ x (1)), . . . , w x (n) · w y (ρ x (n)); ρ y • ρ x ) = (w xy (1), . . . , w xy (n); ρ xy ), which implies that the permutation representation G → S n , x → ρ x is a homomorphism with respect to the opposite law of composition ρ xy = ρ y • ρ x on S n , 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 H n × S n .
A related viewpoint is taken by M. Hall [15, p.200], who uses the multiplication of monomial matrices to describe the wreath product. Such a matrix can be represented in the form M x = diag(w x (1), . . . , w x (n)) · P ρx as the product of an invertible diagonal matrix over the group ring K[H], where K denotes a field, and the permutation matrix P ρx associated with the permutation ρ x ∈ S n . Multiplying two such monomial matrices yields a law of composition identical to the wreath product, M x · M y = diag(w x (1), . . . , w x (n)) · P ρx · diag(w y (1), . . . , w y (n)) · P ρy = diag(w x (1) · w y (ρ x (1)), . . . , w x (n) · w y (ρ x (n))) · P ρx•ρy , in the right-sided variant.
Whereas B. Huppert [18, p.413] uses the monomial representation for defining the Artin transfer by composition with the unsigned determinant, we prefer to give the immediate Definition 3.1 and to merely illustrate the homomorphism property of the Artin transfer with the aid of the monomial representation.   This can be seen in the following manner. [15,Thm.1.5.3,p.12], [18, Satz 2.6, p.6].) Given two elements x ∈ G and y ∈ H, there exist unique permutations λ x ∈ S n , and σ y ∈ S m , such that the associated monomials are given by Then, using Corollary 7.3, we have For each pair of subscripts 1 ≤ i ≤ n and 1 ≤ j ≤ m, we put y i := u x (i) ∈ H and obtain Thus, the image of x under the Artin transfer T G,K is given by =T H,K (T G,H (x)).

3.5.
Wreath product of S m and S n . 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 γ x ∈ S n·m , and using the symbolic notation (ℓh) (i,j) := ℓ i h j for all pairs of The preceding proof has shown that k Therefore, the action of the permutation γ x on the set [1, n] × [1, m] is given by γ x (i, j) = (λ x (i), σ ux(i) (j)). The action on the second component j depends on the first component i (via the permutation σ ux(i) ∈ S m ), whereas the action on the first component i is independent of the second component j. Therefore, the permutation γ x ∈ S n·m can be identified with the multiplet (λ x ; σ ux(1) , . . . , σ ux(n) ) ∈ S n × S n m , which will be written in twisted form in the sequel.
The permutations γ x , 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 S m ≀ S n of the symmetric groups S m and S n with respect to {1, . . . , n}, whose underlying set S n m × S n is endowed with the following law of composition in the left-sided variant.
This law reminds of the chain rule D(g •f )(x) = D(g)(f (x))•D(f )(x) for the Fréchet derivative in x ∈ E of the compositum of differentiable functions f : E → F and g : F → G between complete normed spaces.
The above considerations establish a third representation, the stabilizer representation, of the group G in the wreath product S m ≀ S n , 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 G → S m ≀ S n , x → γ x = (σ ux(1) , . . . , σ ux(n) ; λ x ) of the group G in the wreath product S m ≀ S n of symmetric groups is a group homomorphism.
3.6. Cycle decomposition. Let (ℓ 1 , . . . , ℓ n ) be a left transversal of a subgroup H ≤ G of finite index n = (G : H) ≥ 1 in a group G. Suppose the element x ∈ G gives rise to the permutation λ x ∈ S n of the left cosets of H in G such that If the permutation λ x has the decomposition λ x = t j=1 ζ j into pairwise disjoint (and thus commuting) cycles ζ j ∈ S n of lengths f j ≥ 1, which is unique up to the ordering of the cycles, more explicitly, if and t j=1 f j = n, then the image of x ∈ G under the Artin transfer T G,H is given by Proof. The reason for this fact is that we obtain another left transversal of H in G by putting Let us fix a value of 1 ≤ j ≤ t. For 0 ≤ k ≤ f j − 2, we have xg j,k = xx k ℓ j = x k+1 ℓ j = g j,k+1 ∈ g j,k+1 H, resp. u x (j, k) := g −1 j,k+1 xg j,k = 1 ∈ H. However, for k = f j − 1, we obtain The cycle decomposition corresponds to a double coset decomposition G =∪  Then the image of x ∈ G under the Artin transfer T G,H is given by Proof. xH is a cyclic subgroup of order f in G/H, and a left transversal Hence, the formula for the image of x under the Artin transfer T G,H in the previous section takes the particular shape In particular, the inner transfer of an element x ∈ H is given as a symbolic power of H in G as symbolic exponent. The other extreme is the outer transfer of an element x ∈ G \ H which generates G modulo H, that is G = x, H . It is simply an nth power Proof. The inner transfer of an element x ∈ H, whose coset xH = H is the principal set in G/H of order f = 1, is given as the symbolic power Lemma 3.1. All subgroups H ≤ G of a group G which contain the commutator subgroup G ′ are normal subgroups H G.
If H were not a normal subgroup of G, then we had x −1 Hx ⊆ H for some element x ∈ G \ H. This would imply the existence of elements h ∈ H and y ∈ G \ H such that x −1 hx = y, and consequently the commutator Explicit implementations of Artin transfers in the simplest situations are presented in the following section.

Computational implementation
4.1. Abelianization of type (p, p). Let G be a pro-p group with abelianization G/G ′ of elementary abelian type (p, p). Then G has p + 1 maximal subgroups which is defined as the intersection of all maximal subgroups, coincides with the commutator subgroup G ′ = [G, G], since the latter contains all pth powers G ′ ≥ G p , and thus we have be the Artin transfer homomorphism from G to the abelianization of H i . According to Burnside's basis theorem, the group G has generator rank d(G) = 2 and can therefore be generated as G = x, y by two elements x, y such that x p , y p ∈ G ′ . For each of the normal subgroups H i ⊳G, we need a generator h i with respect to G ′ , and a generator Then, for each 1 ≤ i ≤ p + 1, it is possible to implement the inner transfer by 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 It should be pointed out that the complete specification of the Artin transfers T i also requires explicit knowledge of the derived subgroups H ′ i . Since G ′ is a normal subgroup of index p in H i , a certain general reduction is possible by Lem.2.1, p.52], but an explicit pro-p presentation of G must be known for determining generators of G ′ = s 1 , . . . , s n , whence Abelianization of type (p 2 , p). Let G be a pro-p group with abelianization G/G ′ of nonelementary abelian type (p 2 , p). Then G has p + 1 maximal subgroups Figure 1 visualizes this smallest non-trivial example of a multi-layered abelianization G/G ′ [27, i , be the Artin transfer homomorphism from G to the abelianization of H 1,i , resp. H 2,i . Burnside's basis theorem asserts that the group G has generator rank d(G) = 2 and can therefore be generated as G = x, y by two elements x, y such that We begin by considering the first layer of subgroups. For each of the normal subgroups These are the cases where the factor group H 1,i /G ′ is cyclic of order p 2 . However, for the distinguished maximal subgroup H 1,p+1 , for which the factor group H 1,p+1 /G ′ is bicyclic of type (p, p), we need two generators Further, a generator t i of a transversal must be given such that It is convenient to define Then, for each 1 ≤ i ≤ p + 1, we have the inner transfer Now we continue by considering the second layer of subgroups. For each of the normal subgroups Since ord(u i H 2,i ) = 1, but on the other hand ord(t i H 2,i ) = p 2 and ord(w i H 2,i ) = p, for 1 ≤ i ≤ p + 1, with the single exception that ord(t p+1 H 2,p+1 ) = p, we obtain the following expressions for the inner transfer and for the outer transfer (4.14) Again, it should be emphasized that the structure of the derived subgroups H ′ 1,i and H ′ 2,i must be known explicitly to specify the action of the Artin transfers completely.

Transfer targets and kernels
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 G/G ′ by means of recursive applications of the p-group generation algorithm by Newman [32] and O'Brien [33].
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.

Singulets of transfer targets.
Theorem 5.1. Let G and T be groups. Suppose that H = im(ϕ) = ϕ(G) ≤ T is the image of G under a homomorphism ϕ : G → T , and V = ϕ(U ) is the image of an arbitrary subgroup U ≤ G. 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 (2) The restriction ϕ| U : U → V is an epimorphism which induces a unique epimorphism Thus, the abelianization of V , is an epimorphic image of the abelianization of U , namely the quotient of U/U ′ by the kernel ofφ, which is given by (3) Moreover, the mapφ is an isomorphism, and the quotients V /V ′ ≃ U/U ′ are isomorphic, if and only if See Figure 2 for a visualization of this situation.
The statements can be seen in the following manner. The image of the commutator subgroup is given by The homomorphism ϕ can be restricted to an epimorphism ϕ| U : U → ϕ(U ) = V . According to Theorem 7.1, in particular, by the Formulas (7.5) and (7.4) in the appendix, the condition ϕ(U ′ ) = V ′ implies the existence of a uniquely determined epimorphismφ : The Isomorphism Theorem in Formula (7.7) in the appendix shows that V /V ′ ≃ (U/U ′ )/ ker(φ). Furthermore, by the Formulas (7.4) and (7.1), the kernel ofφ is given explicitly by Thus,φ is an isomorphism if and only if ker(ϕ) U ′ ( U ).
Functor of derived quotients. In analogy to section § 7.6 in the appendix, a covariant functor F : ϕ → F (ϕ) =φ can be used to map a morphism ϕ of one category to an induced morphismφ of another category.
In the present situation, we denote by G the category of groups and we define the domain of the functor F as the following category G s . The objects of the category are pairs (G, U ) consisting of a group G and a subgroup U ≤ G, The functor F : G s → A from this category G s to the category A of abelian groups maps a pair (G, U ) ∈ Obj(G s ) to the commutator quotient group F ((G, U )) : and let equality be defined by Corollary 5.1. If both components of the pairs (G, U ), (H, V ) ∈ Obj(G s ) are restricted to Hopfian groups, then the pre-order of transfer targets V /V ′ U/U ′ 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 Hopfian groups, it is due to the implication

5.2.
Singulets of transfer kernels. Suppose that G and T are groups, remains the same and is therefore finite, and the Artin transfer T H,V from H to V /V ′ exists.
(1) The following connections exist between the two Artin transfers: the required condition for the composita of the commutative diagram in Figure 3, and, consequently, the inclusion of the kernels, (2) A sufficient (but not necessary) condition for the equality of the kernels is: See Figure 3 for a visualization of this scenario.
The truth of these statements can be justified in the following way. The first part has been proved in Proposition 2.1 already: x ∈ ker(ϕ). However, if the condition ker(ϕ) U is satisfied, then we are able to conclude that ℓ −1 j ℓ k = ux ∈ U , and thus j = k. Letφ : U/U ′ → V /V ′ be the epimorphism obtained in the manner indicated in the proof of Theorem 5.1 and Formula (5.2). For the image of x ∈ G under the Artin transfer, we obtaiñ Since ϕ(U ′ ) = ϕ(U ) ′ = V ′ , the right hand side equals T H,V (ϕ(x)), provided that (ϕ(ℓ 1 ), . . . , ϕ(ℓ n )) is a left transversal of V in H, which is correct when ker(ϕ) U . This shows that the diagram in Figure 3 is commutative, that is,φ • T G,U = T H,V • ϕ. It also yields the connection between the permutations λ ϕ(x) = λ x and the monomials u ϕ(x) (i) = ϕ(u x (i)) for all 1 ≤ i ≤ n. As a consequence, we obtain the inclusion ϕ(ker(T G,U )) ≤ ker(T H,V ), if ker(ϕ) U . Finally, if ker(ϕ) U ′ , then the previous section has shown thatφ is an isomorphism. Using the inverse isomorphism, we get More explicitly, we have the following chain of equivalences and implications: we certainly have ker(T H,V ) = ϕ(ker(T G,U )) if ker(ϕ) ≤ U ′ , which is, however, not necessary.
Artin transfers as natural transformations. Artin transfers T G,U can be viewed as components of a natural transformation T between two functors ? and F from the following category G f to the usual category G of groups. The objects of the category G f are pairs (G, U ) consisting of a group G and a subgroup U ≤ G of finite index (G : U ) < ∞, (5.14) Obj  The functor F : G f → G from G f to the category G of groups maps a pair (G, U ) ∈ Obj(G f ) to the commutator quotient group F ((G, U )) := U/U ′ ∈ Obj(G) of the subgroup U , and it maps a morphism ϕ ∈ Mor G f ((G, U ), (H, V )) to the induced epimorphism F (ϕ) :=φ ∈ Mor G (U/U ′ , V /V ′ ) of the restriction ϕ| U : U → V . Note that we must abstain here from letting F map into the subcategory A of abelian groups.
The system T of all Artin transfers fulfils the requirements for a natural transformation T : ? → F between these two functors, since we have 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 ker(ϕ) ≤ U , ker(ψ) ≤ V , 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 (G, U ) which share the same first component G as distinct objects in the categories G s , resp. G f , 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 G (f ) of families, which generalizes the category G f , rather than the category G s . However, we have to prepare this definition with a criterion for the compatibility of a system of subgroups with its image under a homomorphism.
Guided by the property ker we define a restricted 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 φ −1 : V → U and φ : U → V.
After this preparation, we are able to specify the new category G (f ) . The objects of the category G (f ) are pairs (G, (U i ) i∈I ) consisting of a group G and the family of all subgroups U i ≤ G with finite index (G : U i ) < ∞, where I denotes a suitable indexing set. Note that G itself is one of the subgroups U i . 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 (G, ) of morphisms consists of epimorphisms ϕ : G → H satisfying ϕ(G) = H, the image conditions ϕ(U i ) = V i , and the kernel conditions ker(ϕ) ≤ U i , which imply the pre-image conditions ϕ −1 (V i ) = U i , for all i ∈ I, briefly written as arrows ϕ : 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. 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 (G, (U i ) i∈I ), (H, (V i ) i∈I ) ∈ Obj(G (f ) ) be two objects of the category G (f ) , where all members of the families (U i ) i∈I and (V i ) i∈I are Hopfian groups. Then (non-strict) precedence of TTTs is defined by for all i ∈ I, and equality of TTTs 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 (G, (U i ) i∈I ), (H, (V i ) i∈I ) ∈ Obj(G (f ) ) be two objects of the category G (f ) , and let ϕ ∈ Mor G (f ) ((G, (U i ) i∈I ), (H, (V i ) i∈I )) be a morphism between these objects. The stable part and the polarized part of the Artin pattern AP(G) of G with respect to ϕ are defined by , for all i ∈ Pol ϕ (G). Note that the precedence of polarized targets is strict as opposed to polarized kernels. 5.4. The Artin pattern on a descendant tree. Before we specialize to the usual kinds of descendant trees of finite p-groups [26, § 4, pp.163-164] we consider an abstract form of a rooted directed tree T , which is characterized by two relations. Firstly, a basic relation π(G) < G between parent and child (also called immediate descendant ), corresponding to a directed edge G → π(G) of the tree, for any vertex G ∈ T \ {R} which is different from the root R of the tree. Secondly, an induced non-strict partial order relation, π n (G) ≤ G for some integer n ≥ 0, between ancestor and descendant, corresponding to a path G = π 0 (G) → π 1 (G) → · · · → π n (G) of directed edges, for an arbitrary vertex G ∈ T , that is, the ancestor π n (G) is an iterated parent of the descendant. Note that only an empty path with n = 0 starts from the root R of the tree, which has no parent. A brief justification of the partial order: Reflexivity is due to the relation G = π 0 (G). Transitivity follows from the rule π m (π n (G)) = π m+n (G). Antisymmetry is a consequence of the absence of cycles, that is, H = π m (G) = π m (π n (H)) = π m+n (H) implies m = n = 0 and thus H = G. The category of a tree. Now let T be a rooted directed tree whose vertices are groups G ∈ Obj(G). Then we define G T , the category associated with T , as a subcategory of the category G (f ) which was introduced in the Formulas (5.23) and (5.24).
The objects of the category G T are those pairs (G, (U i ) i∈I ) in the object class of the category G (f ) whose first component is a vertex of the tree T , The morphisms of the category G T are selected along the paths of the tree T only. For two objects (G, (U i ) i∈I ), (H, (V i ) i∈I ) ∈ Obj(G T ), the set Mor GT ((G, (U i ) i∈I ), (H, (V i ) i∈I )) of morphisms is either empty or consists only of a single element, In the case of an ancestor-descendant relation between H and G, the specification of the supercategory G (f ) enforces the following constraints on the unique morphism π n : the image relations π n (U i ) = V i and the kernel relations ker(π n ) ≤ U i , for all i ∈ I.
At this position, we must start to be more concrete. In the descendant tree T = T (R) of a group R, which is the root of the tree, the formal parent operator π gets a second meaning as a natural projection π : G → π(G) = G/N , x → π(x) = xN , from the child G to its parent π(G), which is always the quotient of G by a suitable normal subgroup N G. To be precise, the epimorphism π = π G with kernel ker(π) = N is actually dependent on its domain G. Therefore, the formal power π n is only a convenient abbreviation for the compositum π π n−1 (G) • · · · • π π 2 (G) • π π(G) • π G .
As described in [26], there are several possible selections of the normal subgroup N in the parent definition G/N . 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 T (R) is the descendant tree of a finite p-group R, then it is usual to take for N G 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 (τ (G), κ(G)) of all vertices G of a descendant tree T = T (R) 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 (U i ) i∈I in the corresponding object (G, (U i ) i∈I ) of the category G T . The restriction depends on the definition of a parent π(G) in the descendant tree. ( Proof. If parents are defined by π(G) = G/γ c (G) with c = cl(G), then we have ker(π) = γ c (G) and ker(π n ) = γ c+1−n (G) for any 0 ≤ n < c. The largest of these kernels arises for n = c−1. Therefore, uniform comparability of Artin patterns is warranted by the restriction ker(π c−1 ) = γ 2 (G) ≤ U i for all i ∈ I. The parent definition π(G) = G/P c−1 (G) with c = cl p (G) implies ker(π) = P c−1 (G) and ker(π n ) = P c−n (G) for any 0 ≤ n < c. The largest of these kernels arises for n = c − 1. Consequently, a uniform comparability of Artin patterns is guaranteed by the restriction ker(π c−1 ) = P 1 (G) ≤ U i for all i ∈ I.
Finally, in the case of the parent definition π(G) = G/G (d−1) with d = dl(G), we have ker(π) = G (d−1) and ker(π n ) = G (d−n) for any 0 ≤ n < d. The largest of these kernels arises for n = d − 1. Consequently, a uniform comparability of Artin patterns is guaranteed by the condition ker(π d−1 ) = G (1) ≤ U i for all i ∈ I.
Note that the first and third condition coincide since both, G (1) and γ 2 (G), denote the commutator subgroup G ′ . So the family (U i ) i∈I is restricted to the normal subgroups which contain G ′ , as announced in the paragraph preceding Lemma 3.1.
The second condition restricts the family (U i ) i∈I to the maximal subgroups of G inclusively the group G itself and the Frattini subgroup P 1 (G) = Φ(G).
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 G (h) whose objects are subject to more severe conditions than those in Formula (5.23), 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 (G, (U i ) i∈I ) ∈ Obj(G (h) ) be an object of the category G (h) . The Artin pattern, more precisely the restricted Artin pattern, of G is the pair The following Main Theorem shows that any non-metabelian group G with derived length dl(G) ≥ 3 and finite abelianization G/G ′ shares its Artin transfer pattern AP r (G) = (τ (G), κ(G)), in the restricted sense, with its metabelianization, that is the second derived quotient G/G ′′ .
Theorem 5.4. (Main Theorem.) Let G be a (non-metabelian) group with finite abelianization G/G ′ , and denote by G (n) , n ≥ 0, the terms of the derived series of G, that is G (0) := G and G (n+1) := [G (n) , G (n) ] for n ≥ 0, in particular, G (1) = G ′ and G (2) = G ′′ , then (1) every subgroup U ≤ G which contains the commutator subgroup G ′ is a normal subgroup U G of finite index 1 ≤ m := (G : U ) < ∞, (2) for each G ′ U G, there is a chain of normal subgroups Proof. We use the natural epimorphism ω :   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 ℓ p (K) ≥ 2 of the p-class tower F ∞ p (K), 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 AP r (G), which of several assigned candidates G with distinct derived lengths dl(G) ≥ 2 is the actual p-class tower group Gal(F ∞ p (K)|K). (In contrast, ℓ p (K) = 1 can always be recognized with AP r (G).) This is the point where the complete Artin pattern AP c (G) enters the stage. Most recent investigations by means of iterated IPADs of 2 nd order, whose components are contained in AP c (G), enabled decisions between 2 ≤ ℓ p (K) ≤ 3 in [27,29].
Another successful method is to employ cohomological results by I.R. Shafarevich on the relation rank d 2 (G) = dim Fp H 2 (G, F p ) 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 [11,28].
Important examples for the concepts in § 5 are provided in the following subsections. (p, p). Let G be a p-group with abelianization G/G ′ of elementary abelian type (p, p). Then G has p + 1 maximal subgroups Remark 5.3. For brevity, the TKT is identified with the multiplet (κ(i)) 1≤i≤p+1 , whose integer components are given by

Abelianization of type
Here, we take into consideration that each transfer kernel ker(T i ) must contain the commutator subgroup G ′ of G, since the transfer target H i /H ′ i is abelian. However, the minimal case ker(T i ) = G ′ cannot occur, according to Hilbert's Theorem 94.
Definition 5.8. The orbit κ(G) = κ Sp+1 of any representative κ is an invariant of the p-group G and is called its transfer kernel type, briefly TKT.
Remark 5.4. This definition of κ(G) goes back to the origins of the capitulation theory and was introduced by Scholz and Taussky for p = 3 in 1934 [35]. Several other authors used this original definition and investigated capitulation problems further. In historical order, Chang and Foote in 1980 [12], Heider and Schmithals in 1982 [17], Brink in 1984 [9], Brink and Gold in 1987 [10], Nebelung in 1989 [31], and ourselves in 1991 [21] and in 2012 [23].
In the brief form of the TKT κ(G), the natural order is expressed by i ≺ 0 for 1 ≤ i ≤ p + 1. Let #H 0 (G) := #{1 ≤ i ≤ p + 1 | κ(i) = 0} denote the counter of total transfer kernels ker(T i ) = G, which is an invariant of the group G. In 1980, Chang and Foote [12] proved that, for any odd prime p and for any integer 0 ≤ n ≤ p + 1, there exist metabelian p-groups G having abelianization G/G ′ of type (p, p) such that #H 0 (G) = n. However, for p = 2, there do not exist non-abelian 2-groups G with G/G ′ ≃ (2, 2), such that #H 0 (G) ≥ 2. Such groups must be metabelian of maximal class. Only the elementary abelian 2-group G = C 2 × C 2 has #H 0 (G) = 3.
In the following concrete examples for the counters #H 0 (G), 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 [4,5].  (p 2 , p). Let G be a p-group with abelianization G/G ′ of nonelementary abelian type (p 2 , p). Then G possesses p+1 maximal subgroups

Convention.
Suppose that H 1,p+1 = p+1 j=1 H 2,j is the distinguished maximal subgroup which is the product of all subgroups of index p 2 , and H 2,p+1 = ∩ p+1 j=1 H 1,j is the distinguished subgroup of index p 2 which is the intersection of all maximal subgroups, that is the Frattini subgroup Φ(G) of G. First layer. For each 1 ≤ i ≤ p + 1, let T 1,i : G → H 1,i /H ′ 1,i be the Artin transfer homomorphism from G to the abelianization of H 1,i . Definition 5.9. The family κ 1,H (G) = (ker(T 1,i )) 1≤i≤p+1 is called the first layer transfer kernel type of G with respect to H 1,1 , . . . , H 1,p+1 and H 2,1 , . . . , H 2,p+1 , and is identified with (κ 1 (i)) 1≤i≤p+1 , where Remark 5.5. Here, we observe that each first layer transfer kernel is of exponent p with respect to G ′ and consequently cannot coincide with H 1,j for any 1 ≤ j ≤ p, since H 1,j /G ′ is cyclic of order p 2 , whereas H 1,p+1 /G ′ is bicyclic of type (p, p).

Second layer.
For each 1 ≤ i ≤ p + 1, let T 2,i : G → H 2,i /H ′ 2,i be the Artin transfer homomorphism from G to the abelianization of H 2,i .
Definition 5.11. The orbit κ(G) = κ Sp×Sp of any representative κ = (κ 1 , κ 2 ) is an invariant of the p-group G and is called its transfer kernel type, briefly TKT.

Stabilization and polarization in descendant trees
Theorem 5.4 has proved that it suffices to get an overview of the restricted Artin patterns of metabelian groups G with dl(G) = 2, since groups G of derived length dl(G) ≥ 3 will certainly reveal exactly the same patterns as their metabelianizations G/G ′′ .
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 G/G ′ 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 G ′ , resp. Theorem 6.5 on the entire group G, we first focus on the intermediate layer (τ 1 , κ 1 ) of the maximal subgroups G ′ < U i < G.
a nilpolarization and total stabilization if G is a core group with cyclic last lower central equal to the cyclic first upper central.
Proof. Theorems 5.1 and 5.2 tell us that for detecting whether stabilization occurs from parent π(G) to child G, we have to compare the projection kernel ker(π) with the commutator subgroups U ′ i of the four maximal normal subgroups U i ⊳ G, 1 ≤ i ≤ 4. According to [22, Cor.3.2, p.480] these derived subgroups are given by , provided the generators of G are selected as indicated above. On the other hand, the projection kernel ker(π) = γ m−1 (G) is given by Combining this information with m − 1 = c, we obtain the following results.
Suppose that generators of G = x, y are selected such that x ∈ G \ χ 2 (G), if m ≥ 4, and y ∈ χ 2 (G) \ G ′ . We define the main commutator s 2 = [y, x] ∈ γ 2 (G) and the higher commutators The maximal subgroups U i of G contain the commutator subgroup G ′ of G as a normal subgroup of index p and thus are of the shape U i = g i , G ′ . We define a fixed ordering by g 1 = y and g i = xy i−2 for 2 ≤ i ≤ p + 1.
We proceed in the same way as in the proof of Theorem 6.1 and compare the projection kernel ker(π) with the commutator subgroups U ′ i of the p + 1 maximal normal subgroups U i ⊳ G, 1 ≤ i ≤ p + 1. According to [22,Cor.3.1,p.476] they are given by . . , s m−1 = γ 3 (G) for 2 ≤ i ≤ p + 1. if the generators of G are chosen as indicated previously. The cyclic projection kernel is given uniformly by (6.10) ker(π) = γ m−1 (G) = s m−1 = ζ 1 (G).

Extreme interfaces of p-groups.
Finally, what can be said about the extreme cases of non-abelian p-groups having the smallest possible nilpotency class c = 3 for coclass r ≥ 2 and c = 2 for coclass r = 1? In these particular situations, the answers can be given for arbitrary prime numbers p ≥ 2.
Theorem 6.3. Let G be a metabelian p-group with abelianization G/G ′ of type (p, p).
(1) If G is of coclass cc(G) ≥ 2 and nilpotency class cl(G) = 3, then p ≥ 3 must be odd and the coclass must be cc(G) = 2 exactly. (2) If G is of coclass cc(G) = 1 and nilpotency class cl(G) = 2, then G is an extra special p-group of order p 3 and exponent p or p 2 . 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 G/G ′ ≃ (p, p).
Since the minimal nilpotency class c of a non-abelian group with coclass r ≥ 1 is given by c = r + 1, the case cl(G) = 3 cannot occur for cc(G) ≥ 3.
So we are considering metabelian p-groups G with G/G ′ ≃ (p, p), nilpotency class cl(G) = 3 and coclass cc(G) = 2 for odd p ≥ 3, which form the stem of the isoclinism family Φ 6 in the sense of P. Hall. According to [25,Lem.3.1,p.446], the commutator subgroups U ′ i of the maximal subgroups U i < G are cyclic of degree p, for such a group G ∈ Φ 6 (0). However, the kernel of the parent projection π is the bicyclic group ker(π) = γ 3 (G) of type (p, p) [25, § 3.5, p.445], which cannot be contained in any of the cyclic U ′ i with 1 ≤ i ≤ p+1. (2) According to [22,Cor.3.1,p.476], the commutator subgroups U ′ i of all maximal subgroups U i < G are trivial, for a metabelian p-group G of coclass cc(G) = 1 and nilpotency class m − 1 = c = cl(G) = 2, which implies k = 0. Thus, the kernel of the parent projection ker(π) = γ 2 (G) = s 2 is not contained in any U ′ i = 1. In both cases, the final claim is a consequence of the Theorems 5.1 and 5.2. 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. Proof. All possible kernels ker(π) = γ c (G) > 1, resp. ker(π) = P c−1 (G) > 1, resp. ker(π) = G (d−1) > 1, of the parent projections π 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. Proof. This follows from Theorem 5.1, since even the maximal possible kernel ker(π) = γ 2 (G) = G ′ , resp. ker(π) = G (1) = G ′ , of the parent projections π is contained in the commutator subgroup G ′ 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 T with respect to the lower exponent-p central series, only the p-rank of the abelianization r p (G/G ′ ) of the vertices G ∈ T remains stable.
Proof. Denote by ̺ := r p (G/G ′ ) the p-rank of the abelianization of G. According to Theorem 5.1, the maximal possible kernel ker(π) = P 1 (G) = Φ(G) of the parent projections π is the Frattini subgroup which is contained in all maximal subgroups U i of G. According to Proposition 5.1, the map π induces a bijection between the sets of maximal subgroups of the child G and the parent π(G), whose cardinality is given by p ̺ −1 p−1 . Consequently, we have r p (π(G)/π(G) ′ ) = ̺ = r p (G/G ′ ).

Appendix: Induced homomorphism between quotient groups
Throughout this appendix, let φ : G → H be a homomorphism from a source group (domain) G to a target group (codomain) H. 7.1. Image, pre-image and kernel. First, we recall some basic facts concerning the image and pre-image of normal subgroups and the kernel of the homomorphism φ.
Lemma 7.1. Suppose that U ≤ G and V ≤ H are subgroups, and x, y ∈ G are elements.
(1) If U G is a normal subgroup of G, then its image φ(U ) φ(G) is a normal subgroup of the (total) image im(φ) = φ(G).
G is a normal subgroup of G.
(1) There exists an induced homomorphismφ : • If the induced homomorphismφ of the quotients exists, then it is determined uniquely by φ, and its kernel, image and cokernel are given by In particular,φ is a monomorphism if and only if U = φ −1 (V ). Moreover,φ is an epimorphism if and only if H = V · φ(G).
In particular,φ is certainly an epimorphism if φ is onto.
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 Figure 5.
If the normal subgroup U G in the assumptions of Theorem 7.1 is taken as U := φ −1 (V ), then the induced homomorphismφ exists automatically and is a monomorphism.
Note that Proof.
7.3. Factorization through a quotient. Theorem 7.1 can be used to derive numerous special cases. Usually it suffices to consider the quotient group G/U corresponding to a normal subgroup U of the source group G of the homomorphism φ : G → H 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 [20, p.17].
Corollary 7.1. (Factorization through a quotient) Suppose U G is a normal subgroup of G and ω : G → G/U denotes the natural epimorphism onto the quotient. If U ≤ ker(φ), then there exists a unique homomorphismφ : Moreover, the kernel ofφ is given by ker(φ) = ker(φ)/U .
Again we summarize the criterion in a formula: In this situation the homomorphism φ is said to factor or factorize through the quotient G/U via the canonical projection ω and the induced homomorphismφ. 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 V := 1 is the trivial group. The equivalent conditions for the existence of the induced homomorphismφ are φ(U ) = 1 resp. U ⊆ φ −1 (1) = ker(φ). 3. Note that the well-known isomorphism theorem (sometimes also called homomorphism theorem) is a special case of Corollary 7.1. If we put U := ker(φ) and if we assume that φ is an epimorphism with φ(G) = H, then the induced homomorphismφ : G/ ker(φ) → φ(G) is an isomorphism, since ker(φ) = ker(φ)/U ≃ 1.
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., [ 7.4. Application to series of characteristic subgroups. The normal subgroup U G in the assumptions of Corollary 7.1 can be specialized to various characteristic subgroups of G for which the condition U ⊆ ker(φ) can be expressed differently, namely by invariants of series of characteristic subgroups.
Corollary 7.2. The homomorphism φ : G → H can be factorized through various quotients of G in the following way. Let n be a positive integer and p be a prime number.
The following special case is particularly well known. Here we take the commutator subgroup G ′ of G as our charecteristic subgroup, which can either be viewed as the term γ 2 (G) of the lower central series of G or as the term G (1) of the derived series of G. Corollary 7.3. A homomorphism φ : G → H passes through the derived quotient G/G ′ of its source group G if and only if its image im(φ) = φ(G) is abelian.
Proof. Putting n = 1 in the first statement or n = 2 in the second statement of Corollary 7.2 we obtain the well-known special case that φ passes through the abelianization G/G ′ if and only if φ(G) is abelian, which is equivalent to dl(φ(G)) ≤ 1, and also to cl(φ(G)) ≤ 1.

Application to automorphisms.
Corollary 7.4. (Induced automorphism) Let φ : G → H be an epimorphism of groups, φ(G) = H, and assume that σ ∈ Aut(G) is an automorphism of G.
In the second statement, φ is said to have the kernel invariance property (KIP) with respect to σ. The situation of Corollary 7.4 is visualized in Figure 8. Proof. Since φ is supposed to be an epimorphism, the well-known isomorphism theorem in Remark 7.3 yields a representation of the image H = φ(G) ≃ G/ ker(φ) as a quotient.
Remark 7.4. If ker(φ) is a characteristic subgroup of G, then Corollary 7.4 makes sure that any automorphism σ ∈ Aut(G) induces an automorphismσ ∈ Aut(H), where H = φ(G) ≃ G/ ker(φ). The reason is that, by definition, a characteristic subgroup of G is invariant under any automorphism of G.
We conclude this section with a statement about GI-automorphisms (generator-inverting automorphisms) which have been introduced by Boston Theorem 7.2. (Induced generator-inverting automorphism) Let φ : G → H be an epimorphism of groups with φ(G) = H, and assume that σ ∈ Aut(G) is an automorphism satisfying the KIP σ(ker(φ)) = ker(φ), and thus induces an automorphism σ ∈ Aut(H).
If σ is generator-inverting, that is, thenσ is also generator-inverting, that is,σ(y)H ′ = y −1 H ′ for all y ∈ H.
Finally, combining all these formulas and expressing H ∋ y = φ(x) for a suitable x ∈ G, we see thatσ(xG ′ ) = σ(x)G ′ = x −1 G ′ implies the required relation forσ(yH ′ ) =σ(y)H ′ : Figure 9. Functorial properties of induced homomorphisms H/V 7.6. Functorial properties. The mapping F : φ → F (φ) =φ 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 homomorphismsφ between quotient groups, we define the domain of the functor F as the following category G n . The objects of the category are pairs (G, U ) consisting of a group G and a normal subgroup U G,  The functorial properties, which are visualized in Figure 9, can be expressed in the following form. Firstly, F maps the identity morphism 1 G ∈ Mor Gn ((G, U ), (G, U )) having the trivial property 1 G (U ) = U to the identity homomorphism (7.14) F (1 G ) =1 G = 1 G/U ∈ Mor G (G/U, G/U ), and secondly, F maps the compositum ψ • φ ∈ Mor Gn ((G, U ), (K, W )) of two morphisms φ ∈ Mor Gn ((G, U ), (H, V )) and ψ ∈ Mor Gn ((H, V ), (K, W )), which obviously enjoys the required property (ψ • φ)(U ) = ψ(φ(U )) ≤ ψ(V ) ≤ W, to the compositum