Let $T=\operatorname{PSL}\_n(q)$ with $n$ a prime number. Then the $\mathscr{C}\_3$ subgroup $M=\langle x\rangle{:}\langle\sigma\rangle$ of $T$ is isomorphic to $\mathbb{Z}\_{\frac{q^n-1}{(q-1)(n,q-1)}}{:}\mathbb{Z}\_n$, where $x$ comes from the Singer cycle.


Note that $\sigma$ has a matrix which is a permutation matrix corresponding to $(1,2,\dots,n)$ in $\operatorname{SL}\_n(q)$. It follows that $\langle\sigma\rangle$ is preserved by any outer automorphism of $T$.


My question is: is $\langle x\rangle$ preserved by any outer automorphism of $T$? It is true for the diagonal automorphism and the $\mathscr{C}\_3$ subgroup of $\operatorname{PGL}\_n(q)$ is isomorphic to $\mathbb{Z}\_{\frac{q^n-1}{q-1}}{:}\mathbb{Z}\_n$. How about the field automorphism and the graph automorphism? How can we find a matrix form of $x$ in this case?


That is to say, if $o\le\operatorname{Out}(T)$, then is the $\mathscr{C}\_3$ subgroup of $T.o$ just $M.o$? And how about the case when $T=\operatorname{PSU}\_n(q)$ and $M$ is isomorphic to $\mathbb{Z}\_{\frac{q^n+1}{(q+1)(n,q+1)}}{:}\mathbb{Z}\_n$, where $n$ is still a prime number? In this case we only need to consider the field automorphism.



Is the Singer cycle preserved by field automorphisms and graph automorphisms?
This is true by Proposition 4.3.6.(I) of Kleidman and Liebeck's book "The Subgroup Structure of the Finite Classical Groups", which says that, in all cases for the linear and unitary groups, there is a unique conjugacy colass of maximal ${\mathscr C}\_3$-subgroups.


In fact in your situation it is easy to prove it directly using Zsigmondy's theorem: with a few exceptions, if $q$ is a prime power and $n>1$, then there is a prime $r$ dividing $q^n-1$ that does not divide $q^m-1$ for any $m<n$. So your maximal subgroup is in fact the normalizer of a (cyclic) Sylow $r$-subgroup of ${\rm PSL}(n,q)$, and then the result is clear.


The only exceptions to Zsigmondy's theorem that apply here are with $n=2$ and $q$ a Fermat prime, in which case the groups are the normalizers of Sylow $2$-subgroups.


