Originally Posted by
Swlabr Where is it you are stuck on this question?
Let $\displaystyle x, y \in C_a$. Then you need to check that $\displaystyle xy \in Ca$ and $\displaystyle x^{-1} \in Ca$. This is precisely what you would always do to check something is a subgroup.
Checking that $\displaystyle xy$ is in $\displaystyle C_a$ is almost elementary (substitute $\displaystyle `xy'$ in the above condition for $\displaystyle `x$), but the fact that $\displaystyle x^{-1} \in C_a$ is not immediately obvious. However, just take the condition, and pre- and post-multiply by $\displaystyle x^{-1}$ to get that the condition also holds for $\displaystyle x^{-1}$