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 $\mathrm{diag}(|{\mathbf{C}}_G(x){|}_{{\pi }^{\prime }})$, where $x$ runs over a set of representatives of the $\pi$-classes of $G$. In particular, we prove that $\det C=\prod |{\mathbf{C}}_G(x){|}_{{\pi }^{\prime }}.$

7 pages, 2 tables