The transfer homomorphism and splitting theorems

Abstract

Let G be a group and H ≤ G. The we say G splits over H if there exists a subgroup K ≤ G such that G = HK and H ◠ K =1. If it so happens that in addition K ◁ G then we say G splits normally over H. If the structure of the subgroup H or K is particularly nice, say H is cyclic or maybe abelian, then one can expect that the structure of the whole group G will be nice or at least influenced in some way by the structure of H or K. For instance, it's well known that if G splits normally over H and both H and K are solvable than G is also solvable. A more fundamental example is if G splits normally over H, G/K is abelian, and the commutator subgroup G' ≤ H, then G is abelian. It's this influence on the structure of G that makes it important to determine when a group will split over one of its subgroups. But what conditions placed on G or on H are sufficient in order to ensure that G splits over H? Is it possible to actually characterize whether or not G will split over a subgroup H in terms of group theoretic properties of H or G? In the early 1900's mathematicians such as W. Burnside [1], F.G. Frobenius [2], P. Hall [3], and Schur and H. Zassenhaus [6] made efforts to finding the answers to this question for various subgroups H of various groups G. In this expository paper we chronical the development of their work and offer proofs of the theorems they were able to prove. We do this mainly through the use of special homomorphism called the transfer homomorphism.