Class two product

Definition

The class two product of groups is defined as their varietal product corresponding to the variety of groups of nilpotency class two. It is the class two case of the nilpotent product.

For two groups

Suppose $G_{1}$ and $G_{2}$ are groups. Let $G$ be the external free product $G_{1}*G_{2}$, and identify the groups $G_{1}$ and $G_{2}$ with their respective images in $G$. The class two product of $G_{1}$ and $G_{2}$ can be defined as:

$G/[[G_{1},G_{2}],G]$

Structure

For two groups

The subgroup $[G_{1},G_{2}]$ in the class two product is isomorphic to the exterior product of $G_{1}$ and $G_{2}$ viewed as subgroups inside $G_{1}\times G_{2}$, which in turn is isomorphic to the exterior product of their abelianizations. The quotient group by $[G_{1},G_{2}]$ is isomorphic to the external direct product $G_{1}\times G_{2}$.