Details
| Original language | English |
|---|---|
| Pages (from-to) | 323-340 |
| Number of pages | 18 |
| Journal | Journal of algebra |
| Volume | 562 |
| Early online date | 15 Jul 2020 |
| Publication status | Published - 15 Nov 2020 |
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.
Keywords
- Character table, Defect groups
ASJC Scopus subject areas
- Mathematics(all)
- Algebra and Number Theory
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In: Journal of algebra, Vol. 562, 15.11.2020, p. 323-340.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
T1 - Character tables and defect groups
AU - Sambale, Benjamin
N1 - Funding Information: I thank Gabriel Navarro for stimulating discussions on this paper, Christine Bessenrodt for making me aware of [1] and Gunter Malle for providing [21] . Moreover, I appreciate a very careful reading of an anonymous referee. The work is supported by the German Research Foundation ( SA 2864/1-2 and SA 2864/3-1 ).
PY - 2020/11/15
Y1 - 2020/11/15
N2 - 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.
AB - 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.
KW - Character table
KW - Defect groups
UR - http://www.scopus.com/inward/record.url?scp=85088034188&partnerID=8YFLogxK
U2 - 10.48550/arXiv.2007.04919
DO - 10.48550/arXiv.2007.04919
M3 - Article
AN - SCOPUS:85088034188
VL - 562
SP - 323
EP - 340
JO - Journal of algebra
JF - Journal of algebra
SN - 0021-8693
ER -