Group in which every p-local subgroup is strongly p-solvable
From Groupprops
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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Contents
Definition
Let be a finite group and
be an odd prime number. We say that
is a group in which every p-local subgroup is p-solvable if every p-local subgroup of
(i.e., the normalizer of any nonidentity
-subgroup) is a strongly p-solvable group.
Note that for , 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
, it includes p-solvability along with avoiding SL(2,3) among its subquotients. For
, 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 ![]() |
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(for ![]() |
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 |