How to calculate inverses in the group algebra Is there an algorithm to calculate the inverse of an element in the group algebra?
For example, does the element $(1 2 3) + 2 . (1 2)(3 4)$ in the group algebra $\mathbb{C} S_{4}$ have an inverse, and if so, how do I calculate it?
 A: In general, an element $u = \sum_{g \in G} \alpha_g g$ of the group algebra $\mathbb{C}G$ of a finite group $G$ is invertible if and only if $u\sigma$ is an invertible matrix whenever $\sigma$ is an irreducible matrix representation of $G$ (this only depends on the equivalence type of $\sigma$). This also gives a method to construct $u^{-1}$ explicitly when it exists, since the group algebra is a direct sum of matrix algebras, one for each equivalence type of irreducible matrix representation. For larger groups, this is probably impractical, at least by hand. In the particular example you choose, you can note that the element you consider is invertible if and only if $I + 2(321)(12)(24)$ is invertible ( I have multiplied by the inverse of the invertible element $(123))$. Since $(321)(12)(24)$ is a $3$-cycle and all $3$-cycles are conjugate, this element is invertible if and only if $v = I + 2(123)$ is invertible. Now for any choice of irreducible matrixrepresentation $\sigma,$ note that the eigenvalues of $v\sigma$ have the form $1+ 2 \omega$ for some third root of unity $\omega$ (possibly $1$). Such an expression is never zero, so $v$ is invertible. I omit the explicit calculation of the inverse, which is straightforward.
