Elementary divisors of the Cartan matrix for partial characters

Abstract

Let $\pi$ be a set of primes and let $G$ be a finite $\pi$-separable group. We prove that the Cartan matrix $C$ for the $\pi$-partial characters of $G$ is equivalent over the integers to a matrix $\operatorname{diag}(|\mathbf{C}_G(x)|_{\pi'})$, where $x$ runs over a set of representatives of the $\pi$-classes of $G$. In particular, we prove that $\det C = \prod |\mathbf{C}_G(x)|_{\pi'}.$