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Any complete group (that is, with trivial center and outer automorphism group) is isomorphic to its automorphism group. The inverse is not true, as the dihedral group of order 8 is isomorphic to its automorphism group but isn't complete.

Problem 15.29 in the Kourovka Notebook asks to find another exemple of a p-group, but I was wondering, is any other exemple known? Is there any other finite (or even infinite) group (not necessarily a p-group) which is isomorphic to its automorphism group but isn't complete?

frafour
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    The dihedral group of order $12$ is another (non $p$-group) example. By this link the automorphism group of $D_{2n}$ has order $n\varphi(n)$, where $\varphi$ is the Euler totient function, so there are no other finite dihedral examples. – Jianing Song Aug 25 '23 at 19:16

2 Answers2

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The infinite dihedral group $G=D_{\infty}$ is also isomorphic to its own automorphism group, but is not complete - see the article A Note on Groups with Just-Infinite Automorphism Groups.

For finite groups, not $p$-groups, not complete, it is said here: "Whether there exist other groups isomorphic to their automorphism groups is an open problem". For more results, see the article On a question about automorphisms of finite p-groups by G. Cutolo. See also this MO-question.

Element118
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Dietrich Burde
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The answer to your question for non-$p$ groups is yes, and smallest example besides $D_8$ is $D_{12}$ as noted in a comment above. Thanks to having done some computer searches with GAP for such groups for a related problem I obtained a complete list of such groups of order less than 512 which I posted as a partial answer to this MO question: https://mathoverflow.net/questions/27861/periodic-automorphism-towers/225919#225919

Justin Benfield
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