Character tables and defect groups

[摘要] 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 vertical bar D/D'vertical bar = 4 where D' denotes the commutator subgroup of D. We also propose a new characterization of nilpotent blocks in terms of the character table. (C) 2020 Elsevier Inc. All rights reserved.

[发布日期] 2020-11-15

[关键词] Character table;Defect groups