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If $G$ is a group, the socle of $G$, denoted $\mathrm{soc}(G)$, is the subgroup generated by all of the minimal normal subgroups of $G$.

That is, it is the join of subgroups $N$ which are normal in $G$ such that $N\neq \langle e \rangle$ and if $H$ is a normal subgroup of $G$ with $H\subseteq N$, then $H=N$ or $H=\langle e \rangle$. If there are no such subgroups $N$, then the socle of $G$ is defined to be $\langle e\rangle$.

If $G$ is a finite solvable group, then $\mathrm{soc}(G)$ is isomorphic to a product of some number of cyclic groups of prime order.

Knowl status:
  • Review status: reviewed
  • Last edited by John Jones on 2022-05-12 20:39:40
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