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]

T

  1. A left transversal of in is an ordered system of representatives for the left cosets of in such that is a disjoint union.
  2. Similarly, a right transversal of in is an ordered system of representatives for the right cosets of in such that is a disjoint union.

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]

  1. If is non-abelian and is not a normal subgroup of , then we can only say that the inverse elements of a left transversal form a right transversal of in .
  2. However, if is a normal subgroup of , then any left transversal is also a right transversal of in .

For the proof click show on the right hand side.

Let be a group homomorphism and be a left transversal of a subgroup in with finite index . We must check whether the image of this transversal under the homomorphism is again a transversal.

Proposition. 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:

and .

For the proof click show on the right hand side.

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

, for each .

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

, for each .

The elements , resp. , , of the subgroup are called the monomials associated with with respect to , resp. .

Definitions. [1]

The mapping , resp. , is called the permutation representation of in the symmetric group with respect to , resp. .

The mapping , resp. , is called the monomial representation of in with respect to , resp. .

Lemma. For the special right transversal associated to the left transversal , we have the following relations between the monomials and permutations corresponding to an element :

for .

For the proof click show on the right hand side.

Artin transfer[edit]

Let be a group and be a subgroup of finite index . Assume that , resp. , is a left, resp. right, transversal of in with associated permutation representation , resp. , such that , resp. , for .

Definitions. [2] [3]

The Artin transfer from to the abelianization of with respect to , resp. , is defined by

,

resp.

,

for .

Remarks. Isaacs [4] calls the mapping , , resp. , 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 , , resp. , 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]

Assume that is another left transversal of in such that .

Proposition. [1] [2] [4] [5] [7] [8] [9] The Artin transfers with respect to and coincide, that is, .

For the proof click show on the right hand side.

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 .

Proposition. The Artin transfers with respect to and coincide, that is, .

For the proof click show on the right hand side.

Artin transfers as homomorphisms[edit]

Let be a left transversal of in .

Theorem. [1] [2] [4] [5] [7] [8] [9] The Artin transfer and the permutation representation are group homomorphisms:

.

For the proof click show on the right hand side.

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 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 reminds of the 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] This law of composition on causes the monomial representation also to be a homomorphism. In fact, it is an injective homomorphism, also called a monomorphism or embedding, in contrast to the permutation representation, which cannot be injective if is infinite or at least of an order bigger than , the factorial.

For the proof click show on the right hand side.

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]

Let be a group with nested subgroups such that the indices , and are finite.

Theorem. [1] [7] Then the Artin transfer is the compositum of the induced transfer and the Artin transfer , that is,

.

For the proof click show on the right hand side.

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

for all .

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 . Suppose the element gives rise to the permutation of the left cosets of in such that , resp. , for each .

Theorem. [1] [3] [4] [5] [8] [9] 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

.

For the proof click show on the right hand side.

The cycle decomposition corresponds to a double coset decomposition of the group modulo the cyclic group and modulo the subgroup . It was actually 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

.

For the proof click show on the right hand side.

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 modulo , that is .

It is simply an th power

.

For the proof click show on the right hand side.

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 its commutator subgroup . For these intermediate groups we have the following lemma.

Lemma. All subgroups of a group which contain the commutator subgroup are normal subgroups .

For the proof click show on the right hand side.

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 . 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 to the abelianization of . According to Burnside's basis theorem, the group has generator rank and can therefore be generated as by two elements such that . For each of the maximal subgroups , which are normal subgroups by the Lemma in the preceding section, 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 (16) in the last Corollary, 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 (18) in the preceding Corollary. 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 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 , resp. , be the Artin transfer homomorphism from to the abelianization of , resp. . Burnside's basis theorem asserts that the group 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

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

and such that .

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

, for , and .

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

, for , and ,

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

, for , and .

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 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 the commutator subgroup and are therefore necessarily normal, enumerated by means of the 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 each , let be the Artin transfer homomorphism from to the abelianization of .

Definition.

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

Remarks.

  • 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.
  • 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 operation of the symmetric group on the set of all mappings from to , 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 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 .

First layer[edit]

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

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 , 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.

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 from to a subgroup of index () is the compositum of the induced transfer from to and the Artin transfer from to , for any intermediate subgroup of index (). There occur two situations:

  • For the subgroups