some properties of Almost simple group

I will be so thankful for any help due to the following questions. First some notation. Almost simple group "ASG" means group G such that $F^{*}(G)$ is simple non-abelian. I only consider ASG such that its generalized Fitting subgroup is of lie type.

We know simple group $S$ of Lie type in characteristic p have the following properties:

1- $F^*(C_S(X))=O_p(C_S(X))$ for every p-subgroup $X$.

2- For every prime r, $r\neq p$ which divides $|S|$, we have $|S|_p>|S|_r$ (There are some exceptions) where by $|S|_p$ I mean the p-part of $S$

My first question: Do ASG's satisfy in above properties or something like that?

My second question: Is there any relation between the order of Sylow p-subgroup of an ASG and the order of p-sylow subgroup of its generalized Fitting subgroup.

• what's $F^*(G)$? – YCor Jan 2 '14 at 19:45

Well, for a general ASG, the symmetric group $S_{n}$ is almost simple for $n > 4,$ and there is usually no prime $p$ for which property 1 above holds. For example, if the $p$-subgroup $X$ fixes more than $4$ points, $C_{S}(X)$ will have an alternating group as a component. It is possible to work out the order of the largest Sylow subgroup of $S_{n}$ or $A_{n}.$ This is often, but not always, the Sylow $2$-subgroup. However, in $A_{9},$ for example, a Sylow $3$-subgroup has order $81$ and a Sylow $2$-subgroup has order $64.$
As for the ASGs of Lie type, the outer automorphism groups of their generalized Fitting subgroups are solvable. This means that for a $p$-subgroup $X$ of simple $S$ which is $F^{*}(A)$ for some ASG $A$, any component of $C_{A}(X)$ would already be contained in $C_{S}(X)$ as $C_{A}(X)/C_{S}(X)$ is solvable. Hence in Lie type characteristic $p$ ASGs groups, we still have $F^{*}(C_{A}(X) ) = F(C_{A}(X))$ for $p$-subgroups $X.$ It is unusual, but there may be examples where $O_{q}(C_{A}(X)) \neq 1$ for some prime $q \neq p,$ but they should be well understood and documented, eg in Lyons Solomon books.
Answer to comment by user44924 : In the discussion of $p$-constraint above, I was considering $X$ as a subgroup of $S.$ If $X$ is a subgroup of $A$ with $Y = X \cap S \neq 1,$ then $C_{A}(X) \leq C_{A}(Y)$ and nothing much changes. However, if $X \cap S = 1$ ( a relatively rare situation, since we are talking of a characteristic $p$ Lie type group admitting an outer automorphism of order $p,$ and for large primes $p,$ this can only come from a field automorphism) things can be different. For example, if $x$ is a field automorphism of order $p$ of $S = {\rm PSL}(n,p^{p}),$ then $C_{S}(x) = {\rm PSL}(n,p),$ so $C_{S}(x)$ does have a component when $p > 3$ for example.
• Thanks Geoff, Is there such a relation $|A|_p>|A|_r$ for almost simple group of lie type in characteristic p? – Hamid Jan 2 '14 at 7:39