Some questions on quasinilpotent groups and related classes

Abstract

In this paper we will prove that if $G$ is a finite group, $X$ a subnormal subgroup of $X{F}^{\ast }(G)$ such that $X{F}^{\ast }(G)$ is quasinilpotent and $Y$ is a quasinilpotent subgroup of $N_G(X)$, then $Y{F}^{\ast }(N_G(X))$ is quasinilpotent if and only if $Y{F}^{\ast }(G)$ is quasinilpotent. Also we will obtain that $F^{\ast }G$ controls its own fusion in $G$ if and only if $G=F^{\ast }G$.