Character tables and defect groups

TL;DR

This paper establishes new links between the character table of a finite group and the structure of its defect groups, including explicit formulas and characterizations for abelian and cyclic cases.

Contribution

It proves that the exponent of the defect group's center is determined by the character table and provides explicit formulas for abelian defect groups.

Findings

Exponent of the center of D is determined by the character table.

D is cyclic iff B contains a large family of irreducible p-conjugate characters.

Character table can determine if |D/D'|=4 and characterize nilpotent blocks.

Abstract

Let B be a block of a finite group G with defect group D. We prove that the exponent of the center of D is determined by the character table of G. In particular, we show that D is cyclic if and only if B contains a "large" family of irreducible p-conjugate characters. More generally, for abelian D we obtain an explicit formula for the exponent of D in terms of character values. In small cases even the isomorphism type of D is determined in this situation. Moreover, it can read off from the character table whether |D/D'|=4 where D' denotes the commutator subgroup of D. We also propose a new characterization of nilpotent blocks in terms of the character table.