Group in which every p-local subgroup is 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 a 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 p-solvable group.

Relation with other properties

Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
N-group This is a group in which every local subgroup is a solvable group. Follows from the fact that p-local implies local and solvable implies p-solvable
group in which every p-local subgroup is strongly p-solvable We replace the p-solvable group condition by a strongly p-solvable group condition. For p \ge 5, the definitions are equivalent.

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
(for p \ge 5): group in which every p-local subgroup is of Glauberman type
(for p \ge 5): group in which the ZJ-functor controls fusion (via group in which every p-local subgroup is of Glauberman type for p)