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:

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.

□

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

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

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:

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. □

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. □

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:

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. □

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

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

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.

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.,

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

□

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.

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.

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

for, and, then the image of under the Artin transfer is given by

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

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).

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

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

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

with the trace element

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

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.

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

Then, for each, 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

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

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

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

Further, a generator of a transversal must be given such that, for each. It is convenient to define

Then, for each, we have the inner transfer

which equals, and the outer transfer

since and.

Now we continue by considering the second layer of subgroups. For each of the normal subgroups , we select a generator

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

Since, but on the other hand and, for, with the single exception that, we obtain the following expressions for the inner transfer

and for the outer transfer

exceptionally

and

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.

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

2) The restriction 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 by the kernel of, which is given by

3) Moreover, the map is an isomorphism, and the quotients are isomorphic, if and only if

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,

For two objects, the set of morphisms consists of epimor- phisms such that and, briefly written as arrows,

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,

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.

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

and let equality be defined by

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 . □

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,

and, consequently, the inclusion of the kernels,

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

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,

For two objects, the set of morphisms consists of epimor- phisms satisfying, , and the additional condition for their kernels, briefly written as arrows,

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.

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.

The system T of all Artin transfers fulfils the requirements for a natural transformation between these two functors, since we have

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

and let equality be defined by

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. □

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

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 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

,

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,

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

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 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

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 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

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 ([3] , §4, pp. 163-164) we consider an abstract form of a rooted directed tree, which is characterized by two relations.

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,

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,

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),

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

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

3) for each, the targets of the transfers and

are equal in the sense of the natural order,

4) for each, the kernels of the transfers and

are equal in the sense of the natural order,

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 2^nd 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.

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

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.

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

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

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.

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

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 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

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

Combining this information with, we obtain the following results.

・ for if, independently of.

・ for if, which implies.

・ but if, , which also implies.