Continuous functions on primal topological spaces induced by group actions

If $G$ is a group acting on a set $X$, then for any $a\in G$, the restriction ${\varphi }_a:X\to X$ of the action to $a$ induces a topology ${\tau }_a$ for $X$, called the primal topology induced by ${\varphi }_a$. First, we obtain a characterization of the normal subgroups in terms of the primal topologies. Later, we prove that some commutative relations among elements on the group $G$ determine the continuity of maps among different primal spaces $(X,{\tau }_{{\varphi }_x})$. In particular, we prove the continuity of some maps when $a,b,q\in G$ satisfy a quantum type relation, $ba=qab$, as is in the quaternion and Heisenberg groups.