Socle over solvable radical

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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 G is a finite group. The socle over solvable radical of G, denoted \operatorname{Soc}^*(G), is defined as the unique subgroup H of G such that H contains the solvable radical \operatorname{Rad}(G) of G and the quotient group H/\operatorname{Rad}(G) is the socle of the quotient group G/\operatorname{Rad}(G).

Facts

If H is the socle over solvable radical of G, then \operatorname{Rad}(H) = \operatorname{Rad}(G) and H/\operatorname{Rad}(H) is a direct product of simple non-abelian groups.