# Omega subgroups are variety-containing in regular p-group

## Contents

## Definition

Suppose is a Regular p-group (?). Then, for any natural number , the subgroup:

is a Variety-containing subgroup (?) of (i.e., a Variety-containing subgroup of group of prime power order (?)): it contains any subgroup of that is in the subvariety of the variety of groups generated by .

In particular, is a Subhomomorph-containing subgroup (?) and a Subisomorph-containing subgroup (?) of .

## Related facts

- Subisomorph-containing implies omega subgroup in group of prime power order
- Omega subgroups not are subisomorph-containing: In particular, the analogous statements break down for groups that are not regular.

## Facts used

## Proof

By fact (1), is precisely the subset of comprising the elements of whose order divides .

Let be the variety of all groups in which the order of every element divides . Note that this is a variety and it contains . Thus, the subvariety generated by is contained in . Further, any element of that is a subgroup of is contained in . Thus, contains all subgroups of in the subvariety it generates.