Suppose that $G$ is a non-abelian simple group and $r$ is a prime number such that $r$ not divide $|G|$. It is a question for me that whether $r$ can divide |Aut($G$)|?
Prime divisor in the Automorphism group
4
$\begingroup$
group-theory
finite-groups
-
0Do you want $G$ to be non-abelian? – 2012-12-24
-
0@jug: Thanks. Yes, $G$ to be non-abelian simple group. – 2012-12-24
-
5Take a look at the [Suzuki groups](http://en.wikipedia.org/wiki/Suzuki_groups), according to wikipedia they can have automorphisms of order $r = 3$. – 2012-12-24
-
0May I ask if we have any conditions on $r$ or not? I mean if we **just** know that $r\mid |G|$ and want to $r\nmid |\text{Aut}(G)|$?? – 2012-12-24
-
1For a list of simple groups together with the order of the outer morphism group (recall $Out(G)=Aut(G)/Inn(G)$, where $Inn(G)$ are the conjugation automorphisms) see wikipedia: http://en.wikipedia.org/wiki/List_of_finite_simple_groups – 2012-12-24
-
2${\rm PSL}_2(32)$ with $r=5$ is an example. – 2012-12-24
-
1Some more examples: $\text{PSL}_2(128)$ for $r=7$, $\text{PSL}_2(243)$ for $r=5$, $\text{PSL}_3(32)$ for $r=5$. – 2012-12-25