The block graph of a finite group

TL;DR

This paper introduces the block graph of a finite group based on principal block intersections, determines its structure for simple groups, and uses it to characterize group properties like solvability and nilpotency.

Contribution

It defines the block graph for finite groups, characterizes it for simple groups, and links it to group solvability and nilpotency criteria.

Findings

Block graphs are complete for all simple groups except J1 and J4.

The Steinberg character's placement in principal blocks is characterized.

Group solvability and nilpotency can be inferred from block intersection properties.

Abstract

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 $p\neq q$ 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 leads to an equivalent condition for the solvability of a finite group. Thus, together with two recent results of Bessenrodt and Zhang, the nilpotency, $p$-nilpotency and solvability of a finite group can be characterized by intersections of principal blocks of some quotient groups.