# Local subgroup of finite group is contained in p-local subgroup for some prime p

From Groupprops

## Statement

Suppose is a finite group and is a local subgroup of , i.e., it is the normalizer of some nontrivial solvable subgroup of . Then, there exists a prime number and a p-local subgroup of such that .

## Proof

### Proof idea

Suppose for a solvable subgroup of . The idea is to find a prime such that (the p-core of ) is nontrivial. Let . We show that , completing the proof.