Socle over solvable radical

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Definition

Suppose $G$ is a finite group. The socle over solvable radical of $G$, denoted $\operatorname {Soc} ^{*}(G)$, is defined as the unique subgroup $H$ of $G$ such that $H$ contains the solvable radical $\operatorname {Rad} (G)$ of $G$ and the quotient group $H/\operatorname {Rad} (G)$ is the socle of the quotient group $G/\operatorname {Rad} (G)$.

Facts

If $H$ is the socle over solvable radical of $G$, then $\operatorname {Rad} (H)=\operatorname {Rad} (G)$ and $H/\operatorname {Rad} (H)$ is a direct product of simple non-abelian groups.