# Omega subgroups of group of prime power order

## Contents

## Definition

Suppose is a finite -group, i.e. a group of prime power order where the prime is . Then, we define:

In other words, it is the subgroup generated by all elements whose order divides .

If the exponent of is , then . However, there may exist smaller for which .

The -subgroups form an ascending chain of subgroups:

The -subgroups may also be studied for a (possibly infinite) p-group. Since every element in a p-group, by definition, has order a power of , the union of the , for all finite , is the whole group . It may still happen that for some finite .

## Subgroup properties satisfied

All the are clearly characteristic subgroups, and in fact, they're all fully characteristic subgroups: any endomorphism of sends each to within . Even more strongly, all the s are homomorph-containing subgroups, and for a finite -group, they are thus also isomorph-free subgroups.

`Further information: Omega subgroups are homomorph-containing`

## Subgroup-defining function properties

### Monotonicity

*This subgroup-defining function is monotone, viz the image of any subgroup under this function is contained in the image of the whole group*

If is a subgroup, then .

### Idempotence

This subgroup-defining function is idempotent. In other words, applying this twice to a given group has the same effect as applying it once

Applying twice is equivalent to applying it once. In other words, for any , .