Socle over solvable radical
This article defines a subgroup-defining function, viz., a rule that takes a group and outputs a unique subgroup
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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 .
If is the socle over solvable radical of , then and is a direct product of simple non-abelian groups.