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We know that nonisomorphic groups may have isomorphic automorphism groups. As an example, you can think klein four group and $S_3$ since their automorphism group is isomorphic to $S_3$.

Now,I wonder If $G$ and $H$ are nonisomorphic group with same order then can we say that $\operatorname{Aut}(G)$ is not isomorphic to $\operatorname{Aut}(H)$ or can we find two nonisomorphic groups with same order and their automorphism groups are isomorphic?

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up vote 7 down vote accepted


You can check that Automorphism group of both Dihedral group($D_8$) and Direct product of $Z_2$ and $Z_4$ is Dihedral group($D_8$).

So we have two non-isomorphic groups with order $8$ and their Automorphism groups are the same group.

This is the smallest example of such groups.

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