# Fixed points of the automorphisms of sporadic groups

Sporadic groups have very few outer automorphisms (in fact, $$|\mathrm{Out}(G)|\leqslant2$$), so it is very natural to ask what are the fixed points subgroups. For a group of Lie type (and a suitable outer automorphism) one can get a twisted group or the same group over a subfield, but what about sporadic groups? I consulted several sourced but could not find the answer (or, perhaps, I am just not fluent enough in the language of the Atlas).

• The ATLAS lists the orders of the centralizers in $G$ of all elements in ${\rm Out}(G)$, which provides you with some information. For all but the largest of the sporadic groups, you could compute these subgroups in GAP or Magma. I am not really sure exactly what you want to know about them - perhaps you need to ask a more specific question. – Derek Holt Jan 15 at 14:37

For $$K$$ a sporadic group, there you will find an enumeration of the conjugacy classes of subgroups $$\langle x \rangle$$ of prime order in $$\operatorname{Aut}(K)$$, together with descriptions of the structure of $$N_K(\langle x \rangle)$$. So when $$x$$ is noninner, this gives exactly the fixed subgroup in $$K$$.