# Omega subgroups not are variety-containing

This article gives the statement, and possibly proof, of the fact that for a group, the subgroup obtained by applying a given subgroup-defining function (i.e., omega subgroups of group of prime power order) doesnotalways satisfy a particular subgroup property (i.e., variety-containing subgroup)

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

## Statement

We can have a group of prime power order (i.e., a finite -group) such that the first omega subgroup , defined as:

,

is *not* a variety-containing subgroup (i.e., Variety-containing subgroup of group of prime power order (?)) of : there exists a subgroup of isomorphic to a subgroup of but that is not contained in .

In particular, because of the equivalence of definitions of variety-containing subgroup of finite group, need not be a Subisomorph-containing subgroup (?) and it need not be a Variety-containing subgroup (?).

## Related facts

## Proof

`Further information: wreath product of groups of order p`

Suppose is a wreath product of groups of order p, i.e., is a group of order obtained as the semidirect product of an elementary abelian group of order by a cyclic group of order acting as automorphisms. is isomorphic to the -Sylow subgroup of the symmetric group of degree . In particular, has a cyclic subgroup of order .

Suppose is a cyclic group of order .

Define:

.

Then, where is the subgroup of order in . Consider the subgroup . is isomorphic to a subgroup of order in , which in turn is contained in , but itself is not contained in .