Central group algebra over finite group

Question

I am wondering (after reading the Skolem-Noether theorem) whether a group algebra $\Bbb{k}[G]$ can be central (ie. Its center be equal to $\Bbb{k}$). Obviously, the group would be centerless but apart from that?

Answer 1

If $G$ is a group and $a=\sum_g a_gg\in k[G]$, then $a$ is in the center of $k[G]$ iff the coefficients $a_g$ are constant on conjugacy classes. Indeed, given $h\in G$, the condition that $ha=ah$ says exactly that $a_{h^{-1}g}=a_{gh^{-1}}$ for all $g\in G$. Replacing $g$ with $hg$, this is equivalent to saying $a_g=a_{hgh^{-1}}$ for all $g\in G$.

So in particular, the center of $k[G]$ has a basis consisting of sums of the form $\sum_{g\in C} g$ where $C$ is a finite conjugacy class of $G$. So the dimension of the center is the number of finite conjugacy classes. In particular, the center is $k$ iff all conjugacy classes except the identity class are infinite. If $G$ is finite, this means the center can be $k$ only if $G$ is trivial.