Thomas Weigel (University of Milan)

Friday 29th February, 12.05-12.55pm, Carslaw 373

Finite groups with minimal 1-PIM

Let G be a finite group, and let p be a divisor of the order of G. The dimension of P₁ - the indecomposable projective F[G]-module with the trivial module in its head - is a multiple - say c_p(G) - of the maximal p-power dividing the order of G. If G is p-soluble, this value is 1. However, in general the value of c_p(G) is quite mysterious.

Together with G. Malle we have classified all finite simple groups G and prime numbers p for which c_p(G) equals 1. Further analysis shows that for p in {2,3,5} a finite group G satisfying c_p(G)=1 must be p-soluble.