Let $S$ be a non-abelian finite simple group. Also let $\pi(n)$ be the set of prime divisors of a positive integer $n$. Is it true that $$\big|\pi(|\mathrm{Out}(S)|)-\pi(|S|)\big|<\big|\pi(|S|)\big|?$$ Thanks in advance for your attention.
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4$\begingroup$ Could you say a bit more about the motivation for this question? That is, are there some theoretical reasons for hoping it is true, or some numerical evidence? $\endgroup$– Yemon ChoiCommented Jan 23, 2021 at 4:33
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2$\begingroup$ $Out(S)$ is usually very small. Can you use that? See mathoverflow.net/questions/20831/… and answers there. $\endgroup$– markvsCommented Jan 23, 2021 at 4:47
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2$\begingroup$ You can discount sporadic simple groups, which always have outer automorphism groups of order at most $2$, and always have even order. You can also discount alternating groups ( $A_{6}$ needs a little extra care. So the answer is determined by simple groups of Lie type. The outer automorphisms of such a group are products of graph, diagonal, and field automorphisms. There may be some small exceptions, but it is usually the field automorphisms which have order coprime to the order of the simple group (if any at all do). $\endgroup$– Geoff RobinsonCommented Jan 23, 2021 at 14:45
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1$\begingroup$ @GeoffRobinson: There are no exceptions. Outer diagonal automorphisms are matched by the center of the universal version, so can't be coprime. $\endgroup$– Richard LyonsCommented Jan 23, 2021 at 17:54
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$\begingroup$ Thanks for your comments. I am working on almost simple groups and so need to answer this question. $\endgroup$– ebrahimiCommented Jan 24, 2021 at 7:20
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