Characteristic subgroup

A subgroup of a group $G$ that is invariant under all automorphisms of $G$.

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Examples of characteristic subgroups are the centre of a group, denoted by $Z(G)$, the Fitting subgroup, $F(G)$, the commutator subgroup, $D(G)$, $[G,G]$ or $G^{\prime }$, the Frattini subgroup, $\mathrm{\Phi }(G)$, the socle, $\mathrm{Socl}(G)$, the layer, $E(G)$, and the generalized Fitting subgroup. For a definition of the last two, cf. Fitting subgroup.