# 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 ) N-group | ||||

(for ) 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 |