# Omega subgroups are isomorph-free

From Groupprops

This article gives the statement, and possibly proof, of the fact that for any group, the subgroup obtained by applying a given subgroup-defining function (i.e., omega subgroups of a group of prime power order) always satisfies a particular subgroup property (i.e., isomorph-free subgroup)}

View subgroup property satisfactions for subgroup-defining functions View subgroup property dissatisfactions for subgroup-defining functions

## Statement

### Statement with symbols

Let be a group of prime power order. Then, for any natural number , the subgroup:

is an isomorph-free subgroup.

## Facts used

- Omega subgroups are homomorph-containing
- Homomorph-containing implies isomorph-free for co-Hopfian subgroup

## Proof

The proof follows directly from facts (1) and (2), along with the fact that any finite group is co-Hopfian.