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The Schreier conjecture asserts that $\mathrm{Out}(G)$ is always a solvable group when $G$ is a finite simple group. This result is known to be true as a corollary of the classification of finite simple groups, although no simpler proof is known.

Question:

  • What is the obstruction to have a direct transparent proof of Schreier conjecture?

Mathieu group $M_{11}$, $M_{23}$, $M_{24}$ are sporadic simple groups, whose outer automorphism group is trivial.

Mathieu group $M_{12}$, $M_{22}$ are sporadic simple groups, whose outer automorphism group is $\mathbb{Z}_2$.

Tits group ${}^2F_4(2)′$ is a sporadic simple group, whose outer automorphism group is $\mathbb{Z}_2$.

Monster group $\mathbb{M}$ is the largest sporadic simple group, whose outer automorphism group is trivial.

  • Are there some bounds for the outer automorphism groups of sporadic simple groups? (such as bounded by a finite cyclic group $\mathbb{Z}_n$?)
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    $\begingroup$ Sporadic simple groups have outer automorphism groups of order at most $2$. You can find this in any reference on these groups. $\endgroup$ – verret Aug 17 '18 at 2:23
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    $\begingroup$ Not sure there is a good answer to the first question. If you have some "unknown" simple group, you a priori have no information at all about its outer automorphism group. I don't really know the basis on which Schreier made the conjecture (if indeed he did), apart from empirical observation of the known simple groups. H. Wielandt was possibly the first person to prove theorems about the nature of the outer automorphism group of a general finite simple group without knowing the group itself. $\endgroup$ – Geoff Robinson Aug 17 '18 at 10:28
  • $\begingroup$ Somewhat related question here. $\endgroup$ – Jarek Kuben Aug 18 '18 at 14:10
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    $\begingroup$ It's interesting to note, although probably not directly related to your question, that Schreier's hypothesis is the "only" input from the CFSG needed in Babai's recent breakthrough result about graph isomorphism in quasi-polynomial time; see: arxiv.org/abs/1512.03547 $\endgroup$ – Sam Hopkins Aug 18 '18 at 15:22
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Regarding your second question, for every finite simple group the structure of its outer automorphism group is known.

For the sporadic groups, there is the following 2011 preprint on arXiv:

Richard Lyons, Automorphism groups of sporadic groups. https://arxiv.org/abs/1106.3760

This paper determines the structure of $\operatorname{Out}(G)$ for every sporadic simple group $G$, and in the end you find that $|\operatorname{Out}(G)| \leq 2$. Of course all of this was known a long time before, but the point of the paper is to present these calculations in one place. The original references for this seem to be difficult to find.

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