# Superfluous subgroups or small subgroup

The basic characterization of superfluous subgroups of an abelian group is the following one.

Let $A≤G$. Then $A\ll G$ if and only if $A\subseteq \operatorname{Rad}(G)$ and $A$ has no divisible quotient groups.

Now my question: Is there characterization of superfluous subgroups for non-abelian group?

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For finite groups, yes: $A \subseteq \Phi(G)$. In general, no, it is a little crazy. The "no divisible quotient groups" part just becomes kind of crazy. –  Jack Schmidt Sep 12 '12 at 17:40