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Let $G=PSL(n,q)$ be the projective linear group over $\mathbb{F}_q$ and let $\sigma$ be an outer automorphism of $G$. (The description of outer automorphism group of $PSL(n,q)$ is well-known, see for example Wilson's book, Theorem 3.2, page 50.)

The group $G$ acts on $G$ by $g\cdot h=gh\sigma(g^{-1})$ for all $g,h\in G$. The orbits of this action are called $\sigma$-twisted conjugacy classes.

My question is the following:

Is it possible to compute (in terms of the sizes of the classes of $G$) the sizes of these twisted conjugacy classes?

Possible idea:

Let $g\in G$. The $\sigma$-twisted conjugacy class of $g$ can be realized as a conjugacy class of the semidirect product $G\rtimes\langle\sigma\rangle$.

In this context, the question would be the following:

Is it possible to compute the sizes of the conjugacy classes of the group $G\rtimes\langle\sigma\rangle$?

The motivation for my question is the following:

Let $p$ be a prime number. Since there are no conjugacy classes of size $2p$ in finite simple groups (see this post), I would like to prove that there are no twisted conjugacy classes of $PSL(n,q)$ of size $2p$.

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The automorphism should be connected to Galois theory, or not? In the theory of base change one compares conjugacy classes with twisted conjugacy classes of groups for field extensions. – Marc Palm Feb 26 '12 at 17:14
Thanks! I will try to learn something about this connection. – Leandro Vendramin Feb 26 '12 at 20:39
up vote 2 down vote accepted

My inclination at first is to be skeptical: Is there any numerical evidence?. The setting of the question is perhaps nonstandard, since for finite groups of Lie type the starting point for this kind of twisting has more often been the ambient algebraic group. Much of this is influenced by papers and lecture notes of Springer and Steinberg, where graph automorphisms get combined with Frobenius maps. (For instance, there is a more recent note by Springer "Twisted conjugacy in simply connected groups" in Transformation Groups, 2006, which builds on some of the older work.)

Aside from that, not enough solid motivation for the question has been provided, since "twisted" conjugacy classes aren't at first sight all that close to the usual conjugacy classes. To compute sizes of orbits under a finite group action, the typical method is to consider the order of the isotropy groups. Here I'm not sure what is true, but at the least it's important to work through the simplest case $n=3$ in detail. Here the classes of the simple groups in question are well known: see for instance the 1973 paper by Simpson-Frame in Canad. J. Math., where classes and characters are worked out concretely. But it's less clear how to sort out the isotropy groups of the "twisted" conjugacy classes.

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