The Transfer Homomorphsim

The transfer homomorphism from a group G to an abelian subgroup H of finite index in G arises from the natural action of G on the cosets of H. It plays an important role in the study of unsolvable groups and aids in the characterization of groups in which all Sylow p-subgroups are cyclic. In this paper, we will define the transfer homomorphism, and then investigate a few of its properties. Let G be a group with an abelian subgroup H of finite index, [G : H] = n. G acts naturally on the right cosets of H (we will assume all of our cosets are right cosets). Consider a coset Hx and g ∈ G, where Hx 7→ Hxg under the action of G. Define a transversal for H in G to be a set of coset representatives. Let T = {t1, t2 . . . tn} be a right transversal for H in G, with index set J = {1, 2, . . . n}. Then G is equal to ∪j∈JHtj and acts on J . We are interested in a homomorphism defined by the action on the cosets when H is not normal in G. However, we cannot use the canonical homomorphism and mod by H. To solve this problem, observe that for g ∈ G, we can