Permutation kernel

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This article defines a subgroup-defining function, viz., a rule that takes a group and outputs a unique subgroup
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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.