Application of transfer maps on Factor Groups

Abstract

We wish to present the proofs of the following two results on transfer maps.

(i) Let G be a finite group and let K be a normal subgroup of order n of G such that G/K is abelian. If G splits over K with H as a complement to K, 𝜏 is the transfer of G into H, ψ is the map from G to H such that ψ(hk) = h, for all h in H, and k in K, and v is the map defined by v(h) = hⁿ, then 𝜏 = ψV.

(ii) Let G be a finite group. If G has an abelian Sylow p-subgroup then p does not divide |G'∩Z(G)|. Hence we show that if all Sylow p-subgroups of G are abelian then G'∩Z(G) = 1. Also, if G/Z(G) is a π group, then G' is also a π group. However, the converse of this result is not true. A counter example will be presented.