# Omega subgroups not are prehomomorph-contained

From Groupprops

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., prehomomorph-contained subgroup)

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

## Statement

It is possible to have a group of prime power order (i.e., a finite -group for some prime number ), a subgroup of , and a surjective homomorphism , such that is not contained in .

Analogous examples can be constructed for for any .

## Proof

### Example for

`Further information: quaternion group, direct product of Q8 and Z2`

Let be the quaternion group of order eight and be the cyclic group of order two. Define:

Let .

Then, which is a Klein four-group. Also, which is again a Klein four-group. Hence, , so there is a surjective homomorphism <mah>\varphi:K \to \Omega_1(P)</math>. However, is not contained in .