# Outer group automorphisms preserving conjugacy classes of pairs of commuting elements

The following problem appears in the study of braided autoequivalences of representation categories of Drinfeld doubles of finite groups.

Let $G$ be a finite group. An outer automorphism $\alpha$ of $G$ is called class preserving if $\alpha$ induces a trivial permutation of the set of conjugacy classes of $G$. Examples of groups having such automorphisms were first constructed by Burnside in 1913 and many more were constructed since then, see, e.g., a survey
http://www.personal.psu.edu/msm344/blogs/psu/research/burnside-involve-FINAL.pdf by Brooksbank and Mizuhara.

I am interested in automorphisms with a more restrictive property. Namely, consider the set $P(G)$ of classes of pairs of commuting elements of $G$ under simultaneous conjugation. That is, $P(G)$ consists of sets of the form $\{(xax^{-1},\,xbx^{-1})\mid x\in G \}$, where $a,b\in G,\, ab=ba$.

Question: Is there an example of a finite group $G$ and an outer automorphism $\alpha$ of $G$ such that $\alpha$ induces a trivial permutation of $P(G)$?

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Because I don't know if I understand completely the question, can you tell me how the following fails: $G=A_5$ and $\alpha$ conjugation by the transposition $(1,2)$. I only ask because commuting elements in $G$ are of the same (prime) order. –  Steve D Sep 24 '13 at 3:32
@SteveD: The elements of order $5$ split into two conjugacy classes in $A_5$, and these are exchanged by $(1,2)$. In particular, $a$ and $a^2$ are conjugate in $S_5$ but not in $A_5$ for $a$ of order $5$. Thus $(1,2)$ moves the sets of the form $\{(xax^{-1},1)\mid x\in G\}$ or $\{(xax^{-1},xa^2x^{-1})\mid x\in G\}$. –  Frieder Ladisch Sep 24 '13 at 10:33