Characters, commutators and centers of Sylow subgroups

Abstract

The character table of a finite group G G determines whether | P : P ′ | = p 2 |P:P’|=p^2 and whether | P : Z ( P ) | = p 2 |P:\mathbf {Z}(P)|=p^2 , where P P is a Sylow p p -subgroup of G G . To prove the latter, we give a detailed classification of those groups in terms of the generalized Fitting subgroup.