# Socle over solvable radical

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. The **socle over solvable radical** of , denoted , is defined as the unique subgroup of such that contains the solvable radical of and the quotient group is the socle of the quotient group .

## Facts

If is the socle over solvable radical of , then and is a direct product of simple non-abelian groups.