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I know every characteristic subgroup is normal subgroup but converse is not true. I find an example in Klien 4- group.

But I have seen on internet that in Q$_8$, each of cyclic subgroup of order 4 is normal (as index 2) but none of these is characteristic.

There are 3 subgroups of order 4 in Q$_8$, how can I show these 3 subgroups are not characteristic?

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There is an automorphism $\phi$ of $Q_8=\{\pm 1,\pm i,\pm j,\pm k\}$ which cyclically permutes $i,j,k$: $$\phi(\pm 1)=\pm 1, \phi(\pm i)=\pm j, \phi(\pm j)=\pm k, \phi(\pm k)=\pm i$$ So $\phi$ does not preserve any of the three (cyclic) order-4 subgroups $\{1,i,-1,-i\},\{1,j,-1,-j\}, \{1,k,-1,-k\}$ of $Q_8$.

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  • $\begingroup$ Very good,,, I understand clearly $\endgroup$ – Pradip Jun 9 '19 at 6:00

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