Orthogonal groups

WARNING: POTENTIAL TERMINOLOGICAL CONFUSION: Please don't confuse this with orthogonal group

Definition

Suppose $G$ and $H$ are groups. We say that $G$ and $H$ are orthogonal groups if the tensor product of the Abelianizations of $G$ and $H$ is trivial, in other words:

$G/G'\otimes H/H'=0$.