Let $T_1$ and $T_2$ will be disjoint finite sets and let $G_1$, $G_2$ will be respectively certain permutation groups of these sets. Direct sum $G_1 \oplus G_2$ in natural way works on $T_1 \cup T_2$: $$\langle g_1,g_2 \rangle t=\begin{cases}g_1t, t \in T_1\\[2pt]g_2t, t \in T_2\end{cases}$$ Prove that the cyclical index of this activity is expressed by the formula $I_{G_1 \oplus G_2}=I_{G_1}\cdot I_{G_2}$.
I saw similar question at ME but I think that group theory is used in solution to this question.
Unforunately I don't know group theory and I want to use only discrete mathematics knowledge about the Polya theory for this task.
However I don't have any ideas how to do it.
Can you give me some tips?