Equal marginal subgroups for a subvariety not implies equal verbal subgroups

From Groupprops
Jump to: navigation, search

Statement

It is possible to have a group G and two subvarieties \mathcal{V}_1 and \mathcal{V}_2 of the variety of groups such that the following hold:

  • V_1(G) \ne V_2(G), where V_1(G) and V_2(G) denote respectively the \mathcal{V}_1-verbal subgroup and \mathcal{V}_2-verbal subgroup of G.
  • V_1^*(G) = V_2^*(G), where V_1^*(G) and V_2^*(G) denote respectively the \mathcal{V}_1-marginal subgroup and \mathcal{V}_2-marginal subgroup of G.

Related facts

Proof

Perfect and centerless

Consider an example where:

Then:

However:

  • V_1^*(G) is the center of G, which is trivial since G is centerless.
  • V_2^*(G) is the trivial subgroup of G.
  • Thus, V_1^*(G) = V_2^*(G).

Example with a finite nilpotent group

Consider the example where:

Then: