# Constrained for a prime divisor implies not simple non-abelian

## Statement

Suppose is a finite group and is prime divisor of the order of . Then, if is a p-constrained group, cannot be a simple non-abelian group.

## Definitions used

### p-constrained group

`Further information: P-constrained group (?)`

Suppose is a finite group and is a -Sylow subgroup. We say that is -constrained if we have:

.

## Facts used

## Proof

**Given**: A finite group , a prime divisor of the order of . is -constrained.

**To prove**: is not a simple non-abelian group.

**Proof**: Since divides the order of , there exists a nontrivial -Sylow subgroup of . By fact (1), is nontrivial, so the centralizer , which contains is nontrivial. Thus, by the definition of -constrained, we have that is nontrivial. Thus, it is a nontrivial normal subgroup of , so we have .

Consider . If is trivial, then is a -group, and hence by fact (1), is a nontrivial normal subgroup, forcing , forcing to be abelian.

On the other hand, if , then does not divide the order of , a contradiction.