Permutation kernel

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Definition

Suppose $G$ is a finite group, $R=\operatorname {Rad} (G)$ is its solvable radical, and $S=\operatorname {Soc} ^{*}(G)$ 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 $\operatorname {PKer} (G)$, is the kernel of the action of $G$ on these factors induced by the action of $G$ on $S/R$ by conjugation. $\operatorname {PKer} (G)$ contains the socle over solvable radical $S$.

The permutation kernel is part of the Babai-Beals filtration of $G$.