Let $G$ be a finite group whose center $Z(G)$ is trivial. Suppose that the group $\text{Out}(G)$ of outer automorphisms is nontrivial.

Question: Does there always exist an $f \in \text{Aut}(G)$ and a $g \in G$ such that $g$ and $f(g)$ do not lie in the same conjugacy class?

This question can be reformulated in the following way. The automorphism group $\text{Aut}(G)$ acts on the set of conjugacy classes of $G$. As the inner automorphism group $\text{Inn}(G)$ acts trivially on this set, the action descends to the quotient $\text{Out}(G)$. With the same assumptions as above, the question then becomes:

Question: Is the action of $\text{Out}(G)$ on the set of conjugacy classes always nontrivial?

Remark: I restrict myself to the case that $G$ is centerless because this case is of main importance to me. I am not sure though whether there is an easy answer or counterexample to the question in the case that we do not assume $G$ to be centerless.

What I've got so far: the question is positive for the symmetric group $S_{n}$ on $n$ letters for all $n \geq 3$. Indeed, only for $n = 6$ there exists an outer automorphism and this automorphism interchanges the conjugacy class of a cycle of cycle type $(ab)$ and the conjugacy class of a cycle of cycle type $(cd)(ef)(gh)$. Here, $a \neq b$ and the $c, d, e, f, g$ and $h$ are distinct.


This question is first asked by Burnside;

Such automorphisms are called as class preserving automorphisms and denoted by $A_c$ or $Aut_c(G)$. As you also notice $Inn(G) \leq A_c\leq Aut(G)$. For some groups two of such groups can be equal. Check this link. If you search as "Class preserving automorphisms", you can find many papers related the this topic.


In this paper,

In $1947$, G.E.Wall[30] constructed examples ... Interestingly his examples contain $2$ group having class preserving outer automorphisms.

  • $\begingroup$ I couldn't immediately locate any examples with $A_c = {\rm Aut}(G)$, which is what the question was asking for. They are mainly concerned with examples in which $A_c$ is bigger than ${\rm Inn}(G)$. Also, most of these examples are $p$-groups, which have nontrivial centres, although a paper by Hertweck describes some Frobenius groups with $A_c \ne {\rm Inn}(G)$. $\endgroup$ – Derek Holt Feb 10 '15 at 20:04
  • $\begingroup$ @DerekHolt: In this paper, it is mentioned that G.E.Wall found such two example.(In introduction part,third paragraph) researchgate.net/publication/… $\endgroup$ – mesel Feb 10 '15 at 22:01
  • $\begingroup$ @DerekHolt: But I could not find the mentioned paper. $\endgroup$ – mesel Feb 10 '15 at 22:07
  • $\begingroup$ Try arxiv.org/abs/1002.1359 It says that Heineken found $p$-groups with $A = {\rm Aut}_c(G)$. But I have still found no references to groups with $Z(G)=1$ and $A = {\rm Aut}_c(G)$. $\endgroup$ – Derek Holt Feb 11 '15 at 8:59
  • $\begingroup$ @mesel: Thank you for making me aware of these kind of automorphisms. The article from G.E. Wall that you mention can be found here. $\endgroup$ – Adam Battelle Feb 11 '15 at 12:40

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