# Group in which every p-local subgroup is 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

View other prime-parametrized group properties | View other group properties

## Contents

## Definition

Let be a finite group and be a 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 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 , 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 ): group in which every p-local subgroup is of Glauberman type | ||||

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