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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 cp(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 cp(G)
is quite mysterious.
Together with G. Malle we have classified all finite simple groups G and prime numbers p for which cp(G) equals 1. Further analysis shows that for p in {2,3,5} a finite group G satisfying cp(G)=1 must be p-soluble. After the seminar we will take the speaker to lunch. See the Algebra Seminar web page for information about other seminars in the series. Anthony Henderson anthonyh@maths.usyd.edu.au. |