I am looking for a group-algebra with a non-trivial solution to $x^2=x$. That is to say, a solution with $x\neq 1$ and $x \neq 0$ where $1$ is the identity.
We have $x\subset \mathbb{C}[G]$ for some group $G$ and want to find some small group such that $x^2=x$ has non-trivial solutions.
Is this possible and if so can you give an example of such a group? A simple group if possible.
Let $G=\{1,g\}$ be cyclic of order $2$. Then $x=\frac{1+g}{2}$ satisfies $$x^2=\frac{1}{4}(1+2g+g^2)=\frac{1}{4}(1+2g+1)=\frac{1+g}{2}=x.$$
Much more generally, it follows from the representation theory of finite groups that for any finite group $G$, $\mathbb{C}[G]$ is isomorphic to a product of matrix rings $M_n(\mathbb{C})$ for various values of $n$, with the number of factors in the product being the number of conjugacy classes in $G$. It follows immediately that if $G$ is any nontrivial finite group then there are nontrivial solutions to $x^2=x$ in $\mathbb{C}[G]$, since you can take an element that is $1$ on some of the factors and $0$ on others.