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
Let be a group and be a subgroup of finite index
Definitions. 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. 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
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. The mappings:
are called the permutation representation of in the symmetric group with respect to and respectively.
Definition. 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
Definitions. 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: