# On the number of irreducible characters in a 3-block with a minimal nonabelian defect group

On the number of irreducible characters in a 3-block with a minimal nonabelian defect group Let B be a 3-block of a finite group G with a defect group D. In this paper, we are mainly concerned with the number of characters in a particular block, so we shall use Isaacs’ approach to block structure. We consider the block B of a group G as a union of two sets, namely a set of irreducible ordinary characters of G having cardinality k(B) and a set of irreducible Brauer characters of G having cardinality l(B). We calculate k(B) and l(B) provided that D is normal in G and $$D \cong \left\langle {x,y,z|x^{3^n } = y^{3^m } = z^3 = \left[ {x,z} \right] = \left[ {y,z} \right] = 1,\left[ {x,y} \right] = z} \right\rangle \left( {n > m \geqslant 2} \right)$$ D ≅ ⟨ x , y , z | x 3 n = y 3 m = z 3 = [ x , z ] = [ y , z ] = 1 , [ x , y ] = z ⟩ ( n > m ⩾ 2 ) . http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Acta Mathematica Sinica, English Series Springer Journals

# On the number of irreducible characters in a 3-block with a minimal nonabelian defect group

, Volume 33 (9) – Jun 15, 2017
8 pages

/lp/springer_journal/on-the-number-of-irreducible-characters-in-a-3-block-with-a-minimal-Hte0akx4Z0
Publisher
Institute of Mathematics, Chinese Academy of Sciences and Chinese Mathematical Society
Copyright © 2017 by Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Chinese Mathematical Society and Springer-Verlag GmbH Germany
Subject
Mathematics; Mathematics, general
ISSN
1439-8516
eISSN
1439-7617
D.O.I.
10.1007/s10114-017-5792-4
Publisher site
See Article on Publisher Site

### Abstract

Let B be a 3-block of a finite group G with a defect group D. In this paper, we are mainly concerned with the number of characters in a particular block, so we shall use Isaacs’ approach to block structure. We consider the block B of a group G as a union of two sets, namely a set of irreducible ordinary characters of G having cardinality k(B) and a set of irreducible Brauer characters of G having cardinality l(B). We calculate k(B) and l(B) provided that D is normal in G and $$D \cong \left\langle {x,y,z|x^{3^n } = y^{3^m } = z^3 = \left[ {x,z} \right] = \left[ {y,z} \right] = 1,\left[ {x,y} \right] = z} \right\rangle \left( {n > m \geqslant 2} \right)$$ D ≅ ⟨ x , y , z | x 3 n = y 3 m = z 3 = [ x , z ] = [ y , z ] = 1 , [ x , y ] = z ⟩ ( n > m ⩾ 2 ) .

### Journal

Acta Mathematica Sinica, English SeriesSpringer Journals

Published: Jun 15, 2017

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