characteristic subgroup

If $(G,*)$ is a group, then $H$ is a characteristic subgroup of $G$ (written $H\operatorname{\,char\,}G$ ) if every automorphism of $G$ maps $H$ to itself. That is, if $f\in{\rm Aut}(G)$ and $h\in H$ then $f(h)\in H$ .

A few properties of characteristic subgroups:

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

Proofs of these properties:

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Consider $H\operatorname{\,char\,}G$ 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 $H\operatorname{\,char\,}G$ .
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Take $K\operatorname{\,char\,}H$ and $H\trianglelefteq G$ , 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 $K\operatorname{\,char\,}H$ and $H\operatorname{\,char\,}G$ , and let $\phi$ be an automorphism of $G$ . Since $H\operatorname{\,char\,}G$ , $\phi[H]=H$ , so $\phi_{H}$ , the restriction of $\phi$ to $H$ is an automorphism of $H$ . Since $K\operatorname{\,char\,}H$ , so $\phi_{H}[K]=K$ . But $\phi_{H}$ is just a restriction of $\phi$ , so $\phi[K]=K$ . Hence $K\operatorname{\,char\,}G$ .

Title	characteristic subgroup
Canonical name	CharacteristicSubgroup
Date of creation	2013-03-22 12:50:56
Last modified on	2013-03-22 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