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Binzhou Xia
Binzhou Xia
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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 splitnon-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 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?

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Binzhou Xia
Binzhou Xia
  • 767
  • 4
  • 15

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 split extension of its socle?