characteristic subgroup
If $(G,*)$ is a group, then $H$ is a characteristic subgroup of $G$ (written $H\mathrm{char}G$) if every automorphism^{} of $G$ maps $H$ to itself. That is, if $f\in \mathrm{Aut}(G)$ and $h\in H$ then $f(h)\in H$.
A few properties of characteristic subgroups:

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If $H\mathrm{char}G$ then $H$ is a normal subgroup^{} of $G$.

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If $G$ has only one subgroup^{} of a given cardinality then that subgroup is characteristic.

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If $K\mathrm{char}H$ and $H\mathrm{\u22b4}G$ then $K\mathrm{\u22b4}G$. (Contrast with normality of subgroups is not transitive.)

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If $K\mathrm{char}H$ and $H\mathrm{char}G$ then $K\mathrm{char}G$.
Proofs of these properties:

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Consider $H\mathrm{char}G$ under the inner automorphisms^{} of $G$. Since every automorphism preserves $H$, in particular every inner automorphism preserves $H$, and therefore $g*h*{g}^{1}\in H$ for any $g\in G$ and $h\in H$. This is precisely the definition of a normal subgroup.

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Suppose $H$ is the only subgroup of $G$ of order $n$. In general, homomorphisms^{} (http://planetmath.org/GroupHomomorphism) take subgroups to subgroups, and of course isomorphisms take subgroups to subgroups of the same order. But since there is only one subgroup of $G$ of order $n$, any automorphism must take $H$ to $H$, and so $H\mathrm{char}G$.

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Take $K\mathrm{char}H$ and $H\mathrm{\u22b4}G$, and consider the inner automorphisms of $G$ (automorphisms of the form $h\mapsto g*h*{g}^{1}$ for some $g\in G$). These all preserve $H$, and so are automorphisms of $H$. But any automorphism of $H$ preserves $K$, so for any $g\in G$ and $k\in K$, $g*k*{g}^{1}\in K$.

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Let $K\mathrm{char}H$ and $H\mathrm{char}G$, and let $\varphi $ be an automorphism of $G$. Since $H\mathrm{char}G$, $\varphi [H]=H$, so ${\varphi}_{H}$, the restriction^{} of $\varphi $ to $H$ is an automorphism of $H$. Since $K\mathrm{char}H$, so ${\varphi}_{H}[K]=K$. But ${\varphi}_{H}$ is just a restriction of $\varphi $, so $\varphi [K]=K$. Hence $K\mathrm{char}G$.
Title  characteristic subgroup 

Canonical name  CharacteristicSubgroup 
Date of creation  20130322 12:50:56 
Last modified on  20130322 12:50:56 
Owner  yark (2760) 
Last modified by  yark (2760) 
Numerical id  13 
Author  yark (2760) 
Entry type  Definition 
Classification  msc 20A05 
Related topic  FullyInvariantSubgroup 
Related topic  NormalSubgroup 
Related topic  SubnormalSubgroup 
Defines  characteristic 