If $\phi\in\mathsf{Aut}(G)$ fixes conjugacy classes and $G$ is finite then do we have $\phi\in\mathsf{Inn}(G)$?

Question

If a group automorphism $\phi:G\to G$ is an inner automorphism then it fixes conjugacy classes in the sense that $\phi(C)=C$ for every conjugacy class $C$.

When I encountered this for the first time (long ago) I wondered whether the converse of this might be true. Only since short I am wiser because someone provided me a nice counterexample. It concerned the permutation group on the set of positive integers.

However he could not give me (yet) a finite group as counterexample.

This gave birth to my question:

If $\phi:G\to G$ is an automorphism that fixes conjugacy classes and moreover group $G$ is finite then can we conclude that $\phi$ is an inner automorphism?

Thank you for taking notice and sorry on forehand if it is a duplicate.

@MatthewTowers That's a quick response. Thank you very much Matthew. — drhab, Oct 28 '22 at 10:10
No, we haven't in general that result. See e.g. here for a historical counterexample: https://math.stackexchange.com/questions/3838188/inner-automorphism-group-as-the-kernel-of-a-homomorphism/3838521?noredirect=1#comment7918217_3838521 — Kan't, Oct 28 '22 at 11:10
Also relevant: https://math.stackexchange.com/questions/3312586 — sTertooy, Oct 28 '22 at 11:28

score 2 · Answer 1 · edited Nov 18 '22 at 07:43

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It is true for finite simple groups. This is a result of Walter Feit and Gary Seitz (1984). Seitz communicated this to me in a letter, but it was formally published afterwards. I needed the result for classifying finite groups whose complex irreducible characters are all primitive, see here (in particular $(6.1)$).

Article: the result of Feit and Seitz appeared in On finite rational groups and related topics, Illinois Jour. of Math., vol. 33 (1989), pages 103-131. It needs the CFSG.

edited Nov 18 '22 at 07:43

Martin Sleziak

56,060

answered Oct 28 '22 at 10:25

Nicky Hekster

52,147

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Thank you for this extra result, Nicky. I had a look at your profile and was pleasantly surprised. In "de zeventiger jaren" I studied at UvA. Moreover every week I go to church in Amstelveen. – drhab Oct 28 '22 at 10:37
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Leuk Henk! Wellicht dat we samen in de collegebanken hebben gezeten dan ... – Nicky Hekster Oct 28 '22 at 10:54

If $\phi\in\mathsf{Aut}(G)$ fixes conjugacy classes and $G$ is finite then do we have $\phi\in\mathsf{Inn}(G)$?

1 Answers1