Varietal product

Definition

Suppose ${\mathcal {V}}$ is a subvariety of the variety of groups that contains the subvariety of abelian groups. Suppose $G_{i},i\in I$ are groups (not necessarily in ${\mathcal {V}}$). The varietal product of these groups with respect to the variety ${\mathcal {V}}$, also called the verbal product with respect to ${\mathcal {V}}$, is defined as follows:

• Suppose $G$ is the external free product of the groups $G_{i}$ with inclusion maps $\varphi _{i}:G_{i}\to G$.
• Let $V(G)$ denote the verbal subgroup of $G$ corresponding to the variety ${\mathcal {V}}$.
• Let $W$ be the normal closure in $G$ of the join of all the commutator subgroups $[\varphi _{i}(G_{i}),\varphi _{j}(G_{j})]$.
• The varietal product is defined as $G/(V(G)\cap W)$.

Particular cases

Subvariety of interest	Corresponding varietal product
all groups	external free product
all abelian groups	restricted external direct product, equivalent to the internal direct product
all nilpotent groups of nilpotency class $\leq c$	nilpotent product for class $c$