# Characteristic subgroup

From Encyclopedia of Mathematics

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 $G'$, the Frattini subgroup, $\Phi(G)$, the socle, $\mathrm{Socl}(G)$, the layer, $E(G)$, and the generalized Fitting subgroup. For a definition of the last two, cf. Fitting subgroup.

**How to Cite This Entry:**

Characteristic subgroup.

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Characteristic_subgroup&oldid=35104

This article was adapted from an original article by O.A. Ivanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article