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I am working on this problem saying: $$ \text{Inn}(D_8)\cong\text{Inn}(Q_8) $$

I really don't know where can I attack this problem. Thank you

$D_8$ is dihedral group and $Q_8$ is quaternon group, both of order 8.

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1 Answer 1

up vote 4 down vote accepted

The inner automorphisms of any group are simply $G/Z(G)$ since the conjugation map coming from any element in the center is trivial. The centers of $D_8$ and $Q_8$ are both isomorphic to $Z/2$. In the case of $D_8$, the center is generated by the rotation by 180 degrees. In $Q_8$, the center is simply $\{1,-1\}$. It is now not difficult to show that in either case, $Inn(D_8) \cong Z/2 \oplus Z/2 \cong Inn(Q_8)$.

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+1 Nice answer. It can be added that since in both cases the center is of order 2 and since the quotient of any group by its center cannot be cyclic non-trivial, this quotient in both cases has to be the Klein group and they both are, thus, isomorphic. –  DonAntonio Nov 23 '12 at 3:48

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