Can a non-split extension of one non-Abelian finite simple group by another exist?
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4$\begingroup$ No. The outer automorphism group of every finite simple group is soluble. Of course, the proof involves the classification. $\endgroup$– Dave BensonCommented Sep 1, 2023 at 19:13
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3$\begingroup$ Of course that result depends on CFSG. $\endgroup$– Derek HoltCommented Sep 1, 2023 at 19:14
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1$\begingroup$ On the other hand, there is quite an interesting non-split extension of the shape $(A_5)^6.A_5$, where $A_5$ is the alternating group of degree five. $\endgroup$– Dave BensonCommented Sep 1, 2023 at 19:32
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3$\begingroup$ If the action of $H$ on $G$ was trivial then the group would be $G \times H$. $\endgroup$– Derek HoltCommented Sep 1, 2023 at 20:44
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4$\begingroup$ The remaining details. Consider an extension $1\to G\to L\to H\to 1$. By the above solvability result, the natural homomorphism $H\to\mathrm{Out}(G)$ is trivial, so the natural homomorphism $L\to\mathrm{Aut}(G)$ maps into $\mathrm{Inn}(G)$. Hence $GC_L(G)=L$ ($C_L(-)$ denoting the centralizer). Since $G$ has a trivial center, $G\cap C_L(G)=\{1\}$. Hence $L=G\times C_L(G)$. $\endgroup$– YCorCommented Sep 2, 2023 at 9:16
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