Question 92
Let $(G, \cdot)$ be a group, and $H \lt G$ be a proper subgroup of $G$. If there are endomorphisms $f, g, h : G \to G$ of the group $G$, such that $f(xy) = g(x)h(y)$ holds for any $x, y \in G \setminus H$, show that:
a) $g = h$;
b) if $G$ is nonabelian, and $H = Z(G)$, then $f = g = h$.
a) $g = h$;
b) if $G$ is nonabelian, and $H = Z(G)$, then $f = g = h$.