# When is an almost simple group a split extension of its socle?

Here an almost simple group is a finite group whose socle (product of all minimal normal subgroups) is a nonabelian simple group. As an extension of its socle, an almost simple group could be split or non-split. For example, there are four groups with socle $L=PSL(2,9)$ other than $L$, i.e. $S_6$, $PGL(2,9)$, $M_{10}$ and $P\Gamma L(2,9)$; the former two are split extensions of $L$ while the latter two are not. Now the question is how to determine all the almost simple groups which is a non-split extension of its socle?

• I can't give you an accurate answer, but my impression is that almost simple groups are split extensions more often than not. The only cases of non-splitting that I have come across are extensions by a product of a field and a diagonal automorphisms of the same order, which is the case for the ${\rm PSL}(2,9)$ example. The same thing happens for ${\rm PSL}(2,q^2)$ for any odd $q$. And you could would get similar nonsplitting for extensions of ${\rm PSL}(3,q^3)$ with $q \equiv 1 \pmod 3$, for example. – Derek Holt Aug 23 '16 at 17:21
• @DerekHolt Thank you Derek. I have replaced "split" with "non-split" in the question. – Binzhou Xia Aug 24 '16 at 2:46