# Equal marginal subgroups for a subvariety not implies equal verbal subgroups

From Groupprops

## Contents

## Statement

It is possible to have a group and two subvarieties and of the variety of groups such that the following hold:

- , where and denote respectively the -verbal subgroup and -verbal subgroup of .
- , where and denote respectively the -marginal subgroup and -marginal subgroup of .

## Related facts

## Proof

### Perfect and centerless

Consider an example where:

- is a centerless group that is not a perfect group (see centerless not implies perfect), for instance, is isomorphic to symmetric group:S3.
- is the variety of abelian groups.
- is the variety comprising the trivial group.

Then:

- is the derived subgroup of . Since is not perfect, .
- .
- Thus, .

However:

- is the center of , which is trivial since is centerless.
- is the trivial subgroup of .
- Thus, .

### Example with a finite nilpotent group

Consider the example where:

- is the nontrivial semidirect product of Z4 and Z4.
- is the variety of abelian groups.
- is the variety of groups of exponent two (along with the trivial group).

Then:

- is derived subgroup of nontrivial semidirect product of Z4 and Z4. In particular, is the derived subgroup of , and is isomorphic to cyclic group:Z2.
- is the Frattini subgroup of , and it coincides with the center of nontrivial semidirect product of Z4 and Z4, which is isomorphic to Klein four-group.
- is the center of , i.e., it is the center of nontrivial semidirect product of Z4 and Z4, and it is isomorphic to the Klein four-group.
- is the socle of (see socle equals Omega-1 of center in nilpotent p-group), and it in fact coincides with the center, since the center is an elementary abelian group.