Finite groups with minimal 1-PIM

Let $$\mathbb F$$ be a field of characteristic $$\ell > 0$$ and let G be a finite group. It is well-known that the dimension of the minimal projective cover $$\Phi_1^G$$ (the so-called 1-PIM) of the trivial left $$\mathbb F[G]$$ -module is a multiple of the $$\ell$$ -part $$|G|_\ell$$ of the order of G. In this note we study finite groups G satisfying $$\dim_{\mathbb F}(\Phi_1^G)=|G|_\ell$$ . In particular, we classify the non-abelian finite simple groups G and primes $$\ell$$ satisfying this identity (Theorem A). As a consequence we show that finite soluble groups are precisely those finite groups which satisfy this identity for all prime numbers $$\ell$$ (Corollary B). 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 $$\ell^\prime$$ -Hall subgroup for G (Theorem C). An important tool in our proofs is the super-multiplicativity of the dimension of the 1-PIM over short exact sequences (Proposition 2.2).