Characteristic subgroups are normal

Prove that characteristic subgroups are normal. Give an example of a normal subgroup that is not characteristic.

Suppose H \leq G is characteristic, and let \varphi_g denote conjugation by g \in G. We know that \varphi_g is an automorphism of G; since H is characteristic, we have gHg^{-1} = \varphi_g[H] = H for all g \in G. Hence H is normal.

Now define \varphi : Q_8 \rightarrow Q_8 by \varphi(i) = i and \varphi(j) = k, and extend homomorphically to all of Q_8. Note that Q_8 = \langle \varphi(i), \varphi(j) \rangle, so that \varphi is surjective; since Q_8 is finite, \varphi is an automorphism.

Recall that every subgroup of Q_8 is normal. Now \varphi[\langle j \rangle] = \langle \varphi(j) \rangle = \langle k \rangle \neq \langle j \rangle; thus \langle j \rangle is normal but not characteristic in Q_8.

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  • Gobi Ree  On December 15, 2011 at 12:32 am

    Another example: Z_p \times 1 \unlhd Z_p \times Z_p is normal(since it is a subgroup of an abelain group) but it is not fixed under the automorphism \varphi(x,y)=(y,x).

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