# Equal verbal subgroups for a subvariety not implies equal marginal 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 perfect group that is not a centerless group (see perfect not implies centerless), for instance, is isomorphic to special linear group:SL(2,5).
- is the variety of abelian groups.
- is the variety comprising the trivial group.

Then:

- is the derived subgroup of . Since is perfect, .
- .
- Thus, .

However:

- is the center of , which is nontrivial since is not centerless.
- is the trivial subgroup of .
- Thus, .

### Example with a finite nilpotent group

Consider the example where:

- is the central product of D8 and Z4.
- is the variety of abelian groups.
- is the variety of groups of exponent two (along with the trivial group).

Then:

- is the derived subgroup of , and is isomorphic to cyclic group:Z2.
- is the Frattini subgroup of , and it coincides with the derived subgroup of .
- is the center of , and it is isomorphic to cyclic group:Z4.
- is the socle of (see socle equals Omega-1 of center in nilpotent p-group), and it in fact coincides with . In particular, it is
*not*the same as .