Difference between revisions of "Characteristic subgroup"
From Encyclopedia of Mathematics
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| − | A subgroup of a group | + | A subgroup of a group $G$ that is invariant under all automorphisms of $G$. |
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| − | Examples of characteristic subgroups are the [[ | + | 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(2)|Frattini subgroup]], $\Phi(G)$, the [[Socle|socle]], $\mathrm{Socl}(G)$, the layer, $E(G)$, and the generalized Fitting subgroup. For a definition of the last two, cf. [[Fitting subgroup]]. |
Revision as of 18:06, 29 November 2014
A subgroup of a group $G$ that is invariant under all automorphisms of $G$.
Comments
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://encyclopediaofmath.org/index.php?title=Characteristic_subgroup&oldid=15360
Characteristic subgroup. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Characteristic_subgroup&oldid=15360
This article was adapted from an original article by O.A. Ivanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article