13 - Transfer and fusion

Summary

If G is a finite group, H ≤ G, and α: H → A is a homomorphism of H into an abelian group A, then it is possible to construct a homomorphism V : G → A from α in a canonical way. V is called the transfer of G into A via α. If we can show there exists g ∈ G – ker(V), then, as G/ker(V) is abelian, g ∉ G⁽¹⁾ the commutator group of G. In particular G is not nonabelian simple.

It is however in general difficult to calculate gV explicitly and decide whether g ∈ ker(V). To do so we need information about the fusion of g in H; that is information about g^G∩H. Hence chapter 13 investigates both the transfer map and techniques for determining the fusion of elements in subgroups of G.

Section 38 contains a proof of Alperin's Fusion Theorem, which says that p-local subgroups control the fusion of p-elements. To be somewhat more precise, if P is a Sylow p-subgroup of G then we can determine when subsets of P are fused in G (i.e. conjugate in G) by inspecting the p-locals H of G with P ∩ H Sylow in H.

Section 39 investigates normal p-complements. A normal p-complement for a finite group G is a normal Hall p′-subgroup of G. Various criteria for the existence of such objects are generated, The most powerful is the Thompson Normal p-Complement Theorem, which is used in the next section to establish the nilpotence of Frobenius kernels.