Consider the group ring $R = (\mathbb{Z}/5\mathbb{Z})S_3$ . Show that the element $\gamma = \bar{2}(1 2) + \bar{2}(13) + \bar{2}(2 3)$ is in the center of the ring.
The ring has $5^6$ elements, so I can't really go through and and prove that $x\gamma = \gamma x$ $\forall x \in R$. I can see that $\bar{2}$ is in the center of $(\mathbb{Z}/5\mathbb{Z})$ but the individual elements (12), (13), and (23) aren't in the center of $S_3$. How can I prove this?
Edit: Can I assume that an element in the center of $(\mathbb{Z}/5\mathbb{Z})$ multiplied by an element in the center of $S_3$ is an element in the center of R?
Hint: It's enough to prove it only for a set of generators of $S_3$.