# P-solvable group

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 **p-solvable group** if it satisfies the following equivalent conditions:

- has a subnormal series where all the quotients are either -groups or have orders relatively prime to .
- is a pi-separable group for . In other words, has a p-series.
- All the composition factors of that are non-abelian do not have dividing their order.

Note that if does *not* divide the order of , is -solvable. Further, a finite group is a finite solvable group if it is -solvable for every prime dividing the order of .

There is a notion of p-length to measure the length of a p-solvable group; briefly, it measures the number of the successive quotient groups that are p-groups in any -series that minimizes this number.

## Relation with other properties

### Stronger properties

### Weaker properties

- p-constrained group:
*For proof of the implication, refer p-solvable implies p-constrained and for proof of its strictness (i.e. the reverse implication being false) refer p-constrained not implies p-solvable*.