Let $G$ be a group. Suppose that $F^{\bullet}G$ is a filtration on $G$. If $q: G \to K$ is a quotient map, then we get an induced filtration of $K$ given by $F^aK:= q(F^aG) = (F^aG)/(F^aG \cap \ker q)$. Suppose that $F^{\bullet}G$ and $T^{\bullet}G$ are two filtrations on $G$. Define $$\mathsf{g...