Product of all elements in finite Abelian group times random group element

Question

I'm just starting a course on group theory and the question in struggling with is as follows:

Let $G$ be a finite Abelian group and let $x\in G$. Proof: $\prod_{g\in G}xg=\prod_{g\in G}g$

I've tried a couple things, since the group is Abelian we have: $\prod_{g\in G}xg=x^{\left|G\right|}\prod_{g\in G}g$. I know for a fact that $x^{\left|G\right|}=e$ however this is only proved using Lagrange's Theorem which is still about 5 weeks away and I can't use right now. So my question is either:

1. Is there another way, that is without Lagrange's Theorem, to prove that $x^{\left|G\right|}=e$?
2. Do you have any tips for solving this question?

Thanks is advance!

Answer 1

If the elementns of $G$ are $g_1,g_2,g_3,\ldots, g_n$, then $xg_1, xg_2, \ldots, xg_n$ are the same elements, just possibly in a permuted order.