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

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