Finite Groups with Minimal 1-pim

Let F be a field of characteristic ` > 0 and let G be a finite group. It is well-known that the dimension of the minimal projective cover Φ1 (the so-called 1PIM) of the trivial left F[G]-module is a multiple of the `-part |G|` of the order of G. In this note we study finite groups G satisfying dimF(Φ1 ) = |G|`. In particular, we classify the non-abelian finite simple groups G and primes ` satisfying this identity (Thm. A). As a consequence we show that finite soluble groups are precisely those finite groups which satisfy this identity for all prime numbers ` (Cor.B). Another consequence is the fact that the validity of this identity for a finite group G and for a small prime number ` ∈ {2, 3, 5} implies the existence of an `′-Hall subgroup for G (Thm. C). An important tool in our proofs is the super-multiplicativity of the dimension of the 1-PIM over short exact sequences (Prop. 2.2).