# Permutation kernel

From Groupprops

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

View a complete list of subgroup-defining functions OR View a complete list of quotient-defining functions

## Definition

Suppose is a finite group, is its solvable radical, and is its socle over solvable radical, i.e., is the socle of . can be expressed uniquely as a direct product of simple non-abelian groups.

The **permutation kernel** of , denoted , is the kernel of the action of on these factors induced by the action of on by conjugation. contains the socle over solvable radical .

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