Group in which every p-local subgroup is strongly p-solvable

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The article defines a property of groups, where the definition may be in terms of a particular prime that serves as parameter
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Definition

Let G be a finite group and p be an odd prime number. We say that G is a group in which every p-local subgroup is p-solvable if every p-local subgroup of G (i.e., the normalizer of any nonidentity p-subgroup) is a strongly p-solvable group.

Note that for p \ge 5, strongly p-solvable is the same as p-solvable, and thus this group property coincides with group in which every p-local subgroup is p-solvable. For p = 3, it includes p-solvability along with avoiding SL(2,3) among its subquotients. For p = 2, the notion is not defined or considered.

Relation with other properties

Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
(for p \ge 5) N-group
(for p \ge 5) p-nilpotent group

Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
group in which every p-local subgroup is p-constrained p-solvable implies p-constrained
group in which every p-local subgroup is p-stable
group in which every p-local subgroup is of Glauberman type
group in which the ZJ-functor controls fusion
group in which the D*-functor controls fusion