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Simple groups contain minimal simple groups. (English) Zbl 0894.20019
If the word ‘minimal’ were to mean ‘minimal with respect to inclusion’, then the result of the title would be a trivial consequence of finiteness. However, the standard meaning is ‘minimal with respect to involvement’, that is the partial order defined by \(H\text{ l.e. }G\) iff \(H\) is a subquotient of \(G\).
Thus to prove the theorem it is necessary and sufficient to prove that every non-Abelian finite simple group is either minimal in the standard sense, or contains a proper non-Abelian simple subgroup.
This is obvious for the alternating groups, and can be readily checked for the sporadic groups and for any chosen finite set of simple groups. Groups of Lie type of Lie rank at least 3 always contain a subgroup \(\text{SL}_2(q)\) of \(\text{PSL}_3(q)\), and with finitely many exceptions at least one of these is simple. The small rank cases are not difficult and can be investigated case-by-case.
While this is an easy consequence of the classification theorem for finite simple groups, one is tempted to wonder whether it is possible to prove it without CFSG – that would indeed be a worthy extension of Thompson’s heroic classification of the minimal simple groups.

20D05 Finite simple groups and their classification
20E32 Simple groups
20E07 Subgroup theorems; subgroup growth
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