For $|G|=m$ and random $x_1, \dots, x_m\in G$, (dis)prove that $\prod x_i$ is uniformly distributed over the elements of $G$.

Question

For $|G|=m$ and random $x_1, \dots, x_m\in G$, (dis)prove that $\prod x_i$ is uniformly distributed over the elements of $G$.

Answer 1

Let $G = \{g_1, g_2, \ldots, g_m\}$, and let $g \in G$. The key is that $G = \{gg_1, gg_2, \ldots, gg_m\}$. First write down an $m \times m$ matrix of all possible products of pairs $g,h \in G$. What do you notice? How can you continue until you have written down all possible products of $m$-tuples?