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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?

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    $\begingroup$ 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. $\endgroup$ – Derek Holt Aug 23 '16 at 17:21
  • $\begingroup$ @DerekHolt Thank you Derek. I have replaced "split" with "non-split" in the question. $\endgroup$ – Binzhou Xia Aug 24 '16 at 2:46
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See the following paper: A. Lucchini, F. Menegazzo, M. Morigi. On the existence of a complement for a finite simple group in its automorphism group. Special issue in honor of Reinhold Baer (1902–1979). Illinois J. Math. 47 (2003), no. 1-2, 395–418. MR2031330

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  • $\begingroup$ For online access to the article, here is a link. projecteuclid.org/euclid.ijm/1258488162 $\endgroup$ – Jim Humphreys Sep 7 '16 at 14:04
  • $\begingroup$ This article only considers when the full automorphism group splits over the simple group. I guess if the full aut group DOES split, then so do all intermediate almost simples, but if it doesn't, then there's more work to be done... $\endgroup$ – Nick Gill Oct 11 '16 at 15:14

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