Finite groups with minimal 1-PIM.
Let \(G\) be a finite group, let \(\ell\) be a prime number and let \(\mathbb{F}\) be a field of characteristic \(\ell\). We denote by \(\Phi^G_1\) the projective cover of the trivial (left) \(\mathbb{F}[G]\)-module \(\mathbb{F}\), that is the unique (up to isomorphism) projective indecomposable \(\mathbb{F}[G]\)-module with socle (and head) isomorphic to the trivial \(\mathbb{F}[G]\)-module \(\mathbb{F}\). This module is also called the 1-PIM of \(\mathbb{F}[G]\). Obviously, \(\Phi^G_1(\mathbb{F})=\Phi^G_1(\mathbb{F}_\ell)\otimes_{\mathbb{F}_\ell}\mathbb{F}\). It is well-known that the \(\mathbb{F}\)-dimension of \(\Phi_1\) is divisible by \(|G|_\ell\), the \(\ell\)-part of \(|G|\). Define \(c_\ell(G):=\dim_\mathbb{F}(\Phi^G_1)/|G|_\ell\). The main purpose of paper under review is to study pairs \((G,\ell)\) for which \(c_\ell(G)=1\), i.e., finite groups \(G\) and prime numbers \(\ell\) for which \(\Phi^G_1\) has minimal \(\mathbb{F}\)-dimension. In particular, the authors classify the non-Abelian finite simple groups \(G\) and primes \(\ell\) satisfying this identity. As a consequence they show that finite soluble groups are precisely those finite groups which satisfy this identity for all prime numbers \(\ell\). Another consequence is the fact that the validity of this identity for a finite group \(G\) and for a small prime number \(\ell\in\{2,3,5\}\) implies the existence of an \(\ell'\)-Hall subgroup for \(G\).
scientific article; zbMATH DE number 5351813
scientific article