Artin transfer (group theory)

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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 mappings arose as group theoretic counterparts of class extension homomorphisms of abelian extensions of algebraic number fields by applying Artin's reciprocity maps to ideal class groups and analyzing the resulting homomorphisms between quotients of Galois groups. However, independently of number theoretic applications, a partial order on the kernels and targets of Artin transfers has recently turned out to be compatible with parent-descendant relations between finite p-groups (with a prime number p), which can be visualized in descendant trees. Therefore, Artin transfers provide a valuable tool for the classification of finite p-groups and for searching and identifying particular groups in descendant trees by looking for patterns defined by the kernels and targets of Artin transfers. These strategies of pattern recognition are useful in purely group theoretic context, as well as for applications in algebraic number theory concerning Galois groups of higher p-class fields and Hilbert p-class field towers.

Transversals of a subgroup[edit]

Let be a group and be a subgroup of finite index

Definitions.[1] A left transversal of in is an ordered system of representatives for the left cosets of in such that

Similarly a right transversal of in is an ordered system of representatives for the right cosets of in such that

Remark. For any transversal of in , there exists a unique subscript such that , resp. . Of course, this element with subscript which represents the principal coset (i.e., the subgroup itself) may be, but need not be, replaced by the neutral element .

Lemma.[2] Let be a non-abelian group with subgroup . Then the inverse elements of a left transversal of in form a right transversal of in . Moreover if is a normal subgroup of , then any left transversal is also a right transversal of in .

Proof. Since the mapping is an involution of we see that:
For a normal subgroup we have for each .

We must check when the image of a transversal under a homomorphism is also a transversal.

Proposition. Let be a group homomorphism and be a left transversal of a subgroup in with finite index The following two conditions are equivalent:

  • is a left transversal of the subgroup in the image with finite index
Proof. As a mapping of sets maps the union to another union:
but weakens the equality for the intersection to a trivial inclusion:
Suppose for some :
then there exists elements such that
Then we have:
Conversely if then there exists such that But the homomorphism maps the disjoint cosets to equal cosets:

Remark. We emphasize the important equivalence of the proposition in a formula:

Permutation representation[edit]

Suppose is a left transversal of a subgroup of finite index in a group . A fixed element gives rise to a unique permutation of the left cosets of in by left multiplication such that:

Using this we define a set of elements called the monomials associated with with respect to :

Similarly, if is a right transversal of in , then a fixed element gives rise to a unique permutation of the right cosets of in by right multiplication such that:

And we define the monomials associated with with respect to :

Definition.[1] The mappings:

are called the permutation representation of in the symmetric group with respect to and respectively.

Definition.[1] The mappings:

are called the monomial representation of in with respect to and respectively.

Lemma. For the 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
This relation simultaneously shows that, for any , the permutation representations and the associated monomials are connected by and for each .

Artin transfer[edit]

Definitions.[2][3] Let be a group and a subgroup of finite index Assume is a left transversal of in with associated permutation representation such that

Similarly let be a right transversal of in with associated permutation representation such that

The Artin transfer with respect to is defined as:

Similarly we define:

Remarks. Isaacs[4] calls the mappings

the pre-transfer from to . The pre-transfer can be composed with a homomorphism from into an abelian group to define a more general version of the transfer from to via , which occurs in the book by Gorenstein.[5]

Taking the natural epimorphism

yields the preceding definition of the Artin transfer in its original form by Schur[2] and by Emil Artin,[3] which has also been dubbed Verlagerung by Hasse.[6] Note that, in general, the pre-transfer is neither independent of the transversal nor a group homomorphism.

Independence of the transversal[edit]

Proposition.[1][2][4][5][7][8][9] The Artin transfers with respect to any two left transversals of in coincide.

Proof

Let and be two left transversals of in . Then there exists a unique permutation such that:

Consequently for all there exists such that For a fixed element , there exists a unique permutation such that we have:

Therefore, the permutation representation of with respect to is given by which yields: Furthermore, for the connection between the two elements:

we have:

Finally since is abelian and and are permutations, the Artin transfer turns out to be independent of the left transversal:

as defined in formula (5).

It is clear that a similar proof shows that:

Proposition. The Artin transfers with respect to any two right transversals of in coincide.

If we show that the Artin transfer with respect to a right transversal coincides with the Artin transfer with respect to a left transversal then the Artin transfer is independent of teh choice of transversals and only depends on and . To this end we have:

Proposition. The Artin transfers with respect to and coincide.

Proof

Using the commutativity of and formula (4), 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 section.

Artin transfers as homomorphisms[edit]

Theorem.[1][2][4][5][7][8][9] Let be a left transversal of in . The Artin transfer

and the permutation representation:

are group homomorphisms:

Proof

Let :

Since is abelian and is a permutation, we can change the order of the factors in the product:

This relation simultaneously shows that the Artin transfer and the permutation representation are homomorphisms.

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

In the last proof, the image of the product turned out to be

,

which is a very peculiar law of composition discussed in more detail in the following section.

The law is reminiscent of crossed homomorphisms in the first cohomology group of a -module , which have the property for .

Wreath product of H and S(n)[edit]

The peculiar structures which arose in the previous section can also be interpreted by endowing the cartesian product with a special law of composition known as the wreath product of the groups and with respect to the set

Definition. For , the wreath product of the associated monomials and permutations is given by

Theorem.[1][7] With this law of composition on the monomial representation

is an injective homomorphism.

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 means the identity permutation. If , for some , then and consequently

Finally, an application of the inverse inner automorphism with yields , as required for injectivity.

Remark. The monomial representation of the theorem stands in contrast to the permutation representation, which cannot be injective if

Remark. Whereas Huppert[1] uses the monomial representation for defining the Artin transfer, we prefer to give the immediate definitions in formulas (5) and (6) and to merely illustrate the homomorphism property of the Artin transfer with the aid of the monomial representation.

Composition of Artin transfers[edit]

Theorem.[1][7] Let be a group with nested subgroups such that and Then the Artin transfer is the compositum of the induced transfer and the Artin transfer , that is:

.
Proof

If is a left transversal of in and is a left transversal of in , that is and , then

is a disjoint left coset decomposition of with respect to .

Given two elements and , there exist unique permutations , and , such that

Then, anticipating the definition of the induced transfer, we have

For each pair of subscripts and , we put , and we obtain

resp.

Therefore, the image of under the Artin transfer is given by

Finally, we want to emphasize the structural peculiarity of the monomial representation

which corresponds to the composite 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 depends on the first component (via the permutation ), whereas the action on the first component is independent of the second component . Therefore, the permutation can be identified with the multiplet

which will be written in twisted form in the next section.

Wreath product of S(m) and S(n)[edit]

The permutations , which arose as second components of the monomial representation

in the previous section, are of a very special kind. They belong to the stabilizer of the natural equipartition of the set into the 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 the set , whose underlying set is endowed with the following law of composition:

This law reminds of the chain rule for the Fréchet derivative in of the compositum of differentiable functions and between complete normed spaces.

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

of the group 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 is infinite. Formula (10) proves the following statement.

Theorem. The stabilizer representation

of the group in the wreath product of symmetric groups is a group homomorphism.

Cycle decomposition[edit]

Let be a left transversal of a subgroup of finite index in a group and be its associated permutation representation.

Theorem.[1][3][4][5][8][9] Suppose the permutation decomposes into pairwise disjoint (and thus commuting) cycles of lengths which is unique up to the ordering of the cycles. More explicitly, suppose

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

Proof

Define for and . This is a left transversal of in since

is a disjoint decomposition of into left cosets of .

Fix a value of . Then:

Define:

Consequently,

The cycle decomposition corresponds to a double coset decomposition of :

It was this cycle decomposition form of the transfer homomorphism which was given by E. Artin in his original 1929 paper.[3]

Transfer to a normal subgroup[edit]

Let be a normal subgroup of finite index in a group . Then we have , for all , and there exists the quotient group of order . For an element , we let denote the order of the coset in , and we let be a left transversal of the subgroup in , where .

Theorem. Then the image of under the Artin transfer is given by:

.
Proof

is a cyclic subgroup of order in , and a left transversal of the subgroup in , where and is the corresponding disjoint left coset decomposition, can be refined to a left transversal with disjoint left coset decomposition:

of in . Hence, the formula for the image of under the Artin transfer in the previous section takes the particular shape

with exponent independent of .

Corollary. In particular, the inner transfer of an element is given as a symbolic power:

with the trace element

of in as symbolic exponent.

The other extreme is the outer transfer of an element which generates , that is .

It is simply an th 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 in as symbolic exponent.

The outer transfer of an element which generates , that is , whence the coset is generator of with order, is given as the th power

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 of Artin transfers from a group to intermediate groups between and . For these intermediate groups we have the following lemma.

Lemma. All subgroups containing the commutator subgroup are normal.

Proof

Let . If were not a normal subgroup of , 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.

Computational implementation[edit]

Abelianization of type (p,p)[edit]

Let be a p-group with abelianization of elementary abelian type . Then has maximal subgroups of index

Lemma. In this particular case, the Frattini subgroup, which is defined as the intersection of all maximal subgroups coincides with the commutator subgroup.

Proof. To see this note that due to the abelian type of the commutator subgroup contains all p-th powers and thus we have .

For each , let be the Artin transfer homomorphism. According to Burnside's basis theorem the group can therefore be generated by two elements such that For each of the maximal subgroups , which are also normal we need a generator with respect to , and a generator of a transversal such that

A convenient selection is given by

Then, for each we use equations (16) and (18) to implement the inner and outer transfers:

,

The reason is that in and

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 in , a certain general reduction is possible by [10] but a presentation of must be known for determining generators of , whence

Abelianization of type (p2,p)[edit]

Let be a p-group with abelianization of non-elementary abelian type . Then has maximal subgroups of index and subgroups of index For each let

be the Artin transfer homomorphisms. Burnside's basis theorem asserts that the group can be generated by two elements such that

We begin by considering the first layer of subgroups. For each of the normal subgroups , we select a generator

such that . 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:

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

Then, for each , we have inner and outer transfers:

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 particularly 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 and outer transfers

exceptionally

Again, it should be emphasized that the structure of the derived subgroups and must be known to specify the action of the Artin transfers completely.

Transfer kernels and targets[edit]

Let be a group with finite abelianization . Suppose that denotes the family of all subgroups which contain and are therefore necessarily normal, enumerated by a finite index set . For each , let be the Artin transfer from to the abelianization .

Definition.[11] The family of normal subgroups is called the transfer kernel type (TKT) of with respect to , and the family of abelianizations (resp. their abelian type invariants) is called the transfer target type (TTT) of with respect to . Both families are also called multiplets whereas a single component will be referred to as a singulet.

Important examples for these concepts are provided in the following two sections.

Abelianization of type (p,p)[edit]

Let be a p-group with abelianization of elementary abelian type . Then has maximal subgroups of index . For let denote the Artin transfer homomorphism.

Definition. The family of normal subgroups is called the transfer kernel type (TKT) of with respect to .

Remark. 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 , since the transfer target is abelian. However, the minimal case cannot occur.

Remark. A renumeration of the maximal subgroups and of the transfers by means of a permutation 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 action of the symmetric group on the set of all mappings from where the extension of the permutation is defined by and formally

Definition. The orbit of any representative is an invariant of the p-group and is called its transfer kernel type, briefly TKT.

Remark. Let denote the counter of total transfer kernels , which is an invariant of the group . In 1980, S. M. Chang and R. Foote[12] proved that, for any odd prime and for any integer , there exist metabelian p-groups having abelianization of type such that . However, for , there do not exist non-abelian -groups with , which must be metabelian of maximal class, such that . Only the elementary abelian -group has . See Figure 5.

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 H. U. Besche, B. Eick and E. A. O'Brien.[13][14]

For , we have

  • for the extra special group of exponent with TKT (Figure 6),
  • for the two groups with TKTs (Figures 8 and 9),
  • for the group with TKT (Figure 4 in the article on descendant trees),
  • for the group with TKT (Figure 6),
  • for the extra special group of exponent with TKT (Figure 6).

Abelianization of type (p2,p)[edit]

Let be a p-group with abelianization of non-elementary abelian type Then possesses maximal subgroups of index and subgroups of index

Assumption. Suppose

is the distinguished maximal subgroup and

is the distinguished subgroup of index which as the intersection of all maximal subgroups, is the Frattini subgroup of .

First layer[edit]

For each , let denote the Artin transfer homomorphism.

Definition. The family is called the first layer transfer kernel type of with respect to and , and is identified with , where

Remark. Here, we observe that each first layer transfer kernel is of exponent with respect to and consequently cannot coincide with for any , since is cyclic of order , whereas is bicyclic of type .

Second layer[edit]

For each , let be the Artin transfer homomorphism from to the abelianization of .

Definition. The family is called the second layer transfer kernel type of with respect to and , and is identified with where

Transfer kernel type[edit]

Combining the information on the two layers, we obtain the (complete) transfer kernel type of the p-group with respect to and .

Remark. The distinguished subgroups and are unique invariants of 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

the relations between and , and and , are given by

Therefore, is another representative of the orbit of under the action:

of the product of two symmetric groups on the set of all pairs of mappings , where the extensions and of a permutation are defined by and , and formally and

Definition. The orbit of any representative is an invariant of the p-group and is called its transfer kernel type, briefly TKT.

Connections between layers[edit]

The Artin transfer is the composition of the induced transfer from to and the Artin transfer

There are two options regarding the intermediate subgroups

  • 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
But even
Furthermore, when with an element of order with respect to , can belong to only if its th 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 .
FactorThroughAbelianization
Figure 1: Factoring through the abelianization.

Inheritance from quotients[edit]

The common feature of all parent-descendant relations between finite p-groups is that the parent is a quotient of the descendant by a suitable normal subgroup Thus, an equivalent definition can be given by selecting an epimorphism with