Let $G$ be an abelian transitive permutation group acting on $\Omega$. Let $\{g_1, \ldots, g_n\}$ be elements of $G$ such that, $\forall i \not = j$, there is no $k \in \mathbb{N}$ such that $g_i = g_j^k$ or $g_j = g_i^k$.

Question: Is it true that, $\displaystyle g_1^{x_1} = \prod_{i=2}^n g_i^{x_i}$ implies $g_1^{x_1} = 1$ (for $x \in \mathbb{N}^n$)?