PlanetMath: characteristic subgroup

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characteristic subgroup

(Definition)

If is a group, then is a characteristic subgroup of (written $H {\mathrm{\,char\,}}G$ ) if every automorphism of maps 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:

If $H{\mathrm{\,char\,}}G$ then is a normal subgroup of .
If has only one subgroup of a given cardinality then that subgroup is characteristic.
If $K{\mathrm{\,char\,}}H$ and $H\trianglelefteq G$ then $K\trianglelefteq G$ . (Contrast with normality of subgroups is not transitive.)
If $K{\mathrm{\,char\,}}H$ and $H{\mathrm{\,char\,}}G$ then $K{\mathrm{\,char\,}}G$ .

Proofs of these properties:

Consider $H{\mathrm{\,char\,}}G$ under the inner automorphisms of . Since every automorphism preserves , in particular every inner automorphism preserves , 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.
Suppose is the only subgroup of of order . In general, homomorphisms take subgroups to subgroups, and of course isomorphisms take subgroups to subgroups of the same order. But since there is only one subgroup of of order , any automorphism must take to , and so $H{\mathrm{\,char\,}}G$ .
Take $K{\mathrm{\,char\,}}H$ and $H\trianglelefteq G$ , and consider the inner automorphisms of (automorphisms of the form $h\mapsto g*h*g^{-1}$ for some $g\in G$ ). These all preserve , and so are automorphisms of . But any automorphism of preserves , so for any $g\in G$ and $k\in K$ , $g*k*g^{-1}\in K$ .
Let $K{\mathrm{\,char\,}}H$ and $H{\mathrm{\,char\,}}G$ , and let $\phi$ be an automorphism of . Since $H{\mathrm{\,char\,}}G$ , $\phi[H]=H$ , so $\phi_H$ , the restriction of $\phi$ to is an automorphism of . Since $K{\mathrm{\,char\,}}H$ , so $\phi_H[K]=K$ . But $\phi_H$ is just a restriction of $\phi$ , so $\phi[K]=K$ . Hence $K{\mathrm{\,char\,}}G$ .