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

## 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:

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

## Proof

### Perfect and centerless

Consider an example where:

Then:

• $V_1(G) = [G,G]$ is the derived subgroup of $G$. Since $G$ is perfect, $V_1(G) = G$.
• $V_2(G) = G$.
• Thus, $V_1(G) = V_2(G)$.

However:

• $V_1^*(G)$ is the center of $G$, which is nontrivial since $G$ is not centerless.
• $V_2^*(G)$ is the trivial subgroup of $G$.
• Thus, $V_1^*(G) \ne V_2^*(G)$.

### Example with a finite nilpotent group

Consider the example where:

Then:

• $V_1(G)$ is the derived subgroup of $G$, and is isomorphic to cyclic group:Z2.
• $V_2(G)$ is the Frattini subgroup of $G$, and it coincides with the derived subgroup of $G$.
• $V_1^*(G)$ is the center of $G$, and it is isomorphic to cyclic group:Z4.
• $V_2^*(G) = \Omega_1(Z(G))$ is the socle of $G$ (see socle equals Omega-1 of center in nilpotent p-group), and it in fact coincides with $V_1(G) = V_2(G)$. In particular, it is not the same as $V_1^*(G)$.