# Collection of groups satisfying a strong normal replacement condition

From Groupprops

## Contents

## Definition

Suppose is a finite collection of finite -groups, i.e., groups of prime power order where the prime is . We say that satisfies a **strong normal replacement condition** if it satisfies the following equivalent conditions:

- For any finite -group that contains a subgroup isomorphic to an element of , contains a normal subgroup ,
*also*isomorphic to an element of , such that is contained in the normal closure of in . - For any finite -group that contains a 2-subnormal subgroup isomorphic to an element of , contains a normal subgroup ,
*also*isomorphic to an element of , such that is contained in the normal closure of in .# For any finite -group and normal subgroup of , if there exists a subgroup of isomorphic to an element of , there exists a subgroup of that is normal in and is isomorphic to an element of .