Finite groups with minimal 1-PIM

Let F be a field of characteristic ℓ > 0 and let G be a finite group. It is well-known that the dimension of the minimal projective cover Φ_1(G) (the so-called 1-PIM) of the trivial left F[G]-module is a multiple of the ℓ -part |G|_ℓ of the order of G. In this note we study finite groups G satisfying dim(Φ_1(G))=|G|_ℓ. In particular, we classify the non-abelian finite simple groups G and primes ℓ 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 ℓ (Corollary B). Another consequence is the fact that the validity of this identity for a finite group G and for a small prime number ℓ ε {2,3,5} implies the existence of an ℓ′ -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).