I know that there are lots of nontrivial groups that are isomorphic to their automorphism groups like $S_3$. Is there any nontrivial group that is isomorphic to its outer automorphism group? Is there any nontrivial finite group isomorphic to both its inner automorphism group and its outer automorphism group? If so, what is the classification of such groups?
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3$\begingroup$ The most interesting fact in this area is that $S_6$ is not isomorphic to its group of automorphisms, while all the other finite symmetric groups are. This is related to a whole host of interesting exceptional cases. $\endgroup$– Mark BennetOct 6, 2022 at 16:07
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5$\begingroup$ I just did a quick computer check, and $\mathtt{SmallGroup}(16,11)$ is an example. This group is the direct produce of a dihedral group of order $8$ and a cyclic group of order $2$. $\endgroup$– Derek HoltOct 6, 2022 at 16:10
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2$\begingroup$ SmallGroup(16, 11) is $C_2 \times D_4$, the product of the cyclic group of order 2 with the dihedral group of order 8. You can see its LMFDB entry here (beta site because abstract groups aren't on the main LMFDB site yet): beta.lmfdb.org/Groups/Abstract/16.11 $\endgroup$– Daniel HastOct 6, 2022 at 16:15
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2$\begingroup$ There is, by the way, even a simple Lie algebra for $p=3$, which is isomorphic to its outer derivation algebra. This is the "Lie algebra version" of your question. The Lie algebra is $\mathfrak{psl}(\Bbb F_3)$. $\endgroup$– Dietrich BurdeOct 6, 2022 at 18:15
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2$\begingroup$ As a general rule you shouldn't edit questions with accepted answers. Ask a new question. There is also a general guideline to ask one question at a time and you often ask several questions at once. But I am very confident that there is no classification of groups with these properties. Why should there be? $\endgroup$– Derek HoltOct 27, 2022 at 19:01
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There is a famous result, that the outer automorphism group of a finite simple group is solvable, see the Schreier conjecture. In particular, it cannot be isomorphic to the group itself.
It is interesting to note that the corresponding question for Lie algebras, posed by Hans Zassenhaus, is different, i.e., there are indeed simple Lie algebras isomorphic to their outer derivation algebra, namely for example the simple Lie algebra $\mathfrak{psl}_3(\Bbb F)$ of dimension $7$ over a field $\Bbb F$ of characteristic three.
For solvable groups $G$, it can easily happen that $G\cong Out(G)$, e.g., for $G=C_2\times D_4$, as noted by Derek. There are some more posts related to this topic, or similar topics, for example
Is every finite group the outer automorphism group of a finite group?
answered Oct 6, 2022 at 18:40