HALL SUBGROUPS AND -COCYCLE REGULARITY

Abstract

AbstractLet H be a subgroup of a finite group G and let $\alpha $ be a complex-valued $2$ -cocycle of $G.$ Conditions are found to ensure there exists a nontrivial element of H that is $\alpha $ -regular in $G.$ However, a new result is established allowing a prime by prime analysis of the Sylow subgroups of $C_G(x)$ to determine the $\alpha $ -regularity of a given $x\in G.$ In particular, this result implies that every $\alpha _H$ -regular element of a normal Hall subgroup H is $\alpha $ -regular in $G.$