We prove that over an algebraically closed field $\mathbb{ K}$ of characteristic different from $2$, the group algebra $R=\mathbb{ K} D_\infty$ of the infinite dihedral group $D_\infty$ has exactly six conjugacy classes of involutions (equivalently, of idempotents). This allows us to recover the fact that $R$ admits exactly four non-isomorphic indecomposable projective modules of the form $eR$ where $e$ is an idempotent, a result that was first established by Berman and Buz\'asi.
Categories: math.GR math.RT