Character tables and defect groups

authored by

Benjamin Sambale

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.

Organisation(s)

Institute of Algebra, Number Theory and Discrete Mathematics

Type

Article

Journal

Journal of algebra

Volume

562

Pages

323-340

No. of pages

18

ISSN

0021-8693

Publication date

15.11.2020

Publication status

Published

Peer reviewed

Yes

ASJC Scopus subject areas

Algebra and Number Theory

Electronic version(s)

https://doi.org/10.48550/arXiv.2007.04919 (Access: Open)
https://doi.org/10.1016/j.jalgebra.2020.05.040 (Access: Closed)