Permutation kernel

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Definition

Suppose $G$ is a finite group, $R=\mathrm{R}\mathrm{a}\mathrm{d}(G)$ is its solvable radical, and $S=\stackrel{\mathrm{o}}{\mathrm{S}}(G)$ is its socle over solvable radical, i.e., $S/R$ is the socle of $G/R$. $S/R$ can be expressed uniquely as a direct product of simple non-abelian groups.

The permutation kernel of $G$, denoted $\mathrm{P}\mathrm{K}\mathrm{e}\mathrm{r}(G)$, is the kernel of the action of $G$ on these factors induced by the action of $G$ on $S/R$ by conjugation. $\mathrm{P}\mathrm{K}\mathrm{e}\mathrm{r}(G)$ contains the socle over solvable radical $S$.