The block graph of a finite group

Julian Brough, Yanjun Liu, Alessandro Paolini

Published 2017-05-24Version 1

This paper studies intersections of principal blocks of a finite group with respect to different primes. We first define the block graph of a finite group G, whose vertices are the prime divisors of |G| and there is an edge between two vertices p \ne q if and only if the principal p- and q-blocks of G have a nontrivial common complex irreducible character of G. Then we determine the block graphs of finite simple groups, which turn out to be complete except those of J_1 and J_4. Also, we determine exactly when the Steinberg character of a finite simple group of Lie type lies in a principal block. Based on the above investigation, we obtain a criterion for the p-solvability of a finite group which in particular asserts that a finite group whose block graph has no triangle containing 2 is solvable.