1. Fachbereich Mathematik, Universit?t Kaiserslautern, Postfach 3049, 67653, Kaiserslautern, Germany 2. Dipartimento di Matematica e Applicazioni, Università di Milano-Bicocca, Via R. Cozzi 53, 20125, Milan, Italy
Abstract:
Let be a field of characteristic and let G be a finite group. It is well-known that the dimension of the minimal projective cover (the so-called 1-PIM) of the trivial left -module is a multiple of the -part of the order of G. In this note we study finite groups G satisfying . In particular, we classify the non-abelian finite simple groups G and primes 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 (Corollary B). Another consequence is the fact that the validity of this identity for a finite group G and for a small prime number implies the existence of an -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).