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 \(D \cong \left\langle {x,y,z|x^{3^n } = y^{3^m } = z^3 = \left[ {x,z} \right] = \left[ {y,z} \right] = 1,\left[ {x,y} \right] = z} \right\rangle \left( {n > m \geqslant 2} \right)\) .
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References
Berkovich, Y.: Finite p-groups with few minimal nonabelian subgroups. J. Algebra, 297, 62–100 (2006)
Broué, M., Puig, L.: A frobenius theorem for blocks. Inventiones Math., 56, 117–128 (1980)
Dade, E. C.: Blocks with cyclic defect groups. Ann. of Math., 84, 20–48 (1966)
Gao, S., Zeng, J. W.: On the number of ordinary irreducible characters in a p-block with a minimal nonabelian defect group. Comm. Algebra, 39(9), 3278–3297
Hendren, S.: Extra special defect groups of order p 3 and exponent p 2. J. Algebra, 291, 457–491 (2005)
Hendren, S.: Extra special defect groups of order p 3 and exponent p. J. Algebra, 313, 724–760 (2007)
Holloway, M., Koshitani, S., Kunugi, N.: Blocks with nonabelian defect groups which have cyclic subgroups of index p. Arch. Math., 94, 101–116 (2010)
Isaacs, I. M.: Character Theory of Finite Groups, Academic Press, Inc., New York, 1976
Kiyota, M.: On 3-blocks with an elementary abelian defect group of order 9. J. Fac. Sci. Univ. Tokyo Sect. IA Math., 31(1), 33–58 (1984)
Sambale, B.: 2-blocks with minimal nonabelian defect groups. J. Algebra, 337, 261–284 (2011)
Xu, X. Z., Liu H. G.: The automorphism group of a finite p-group with derived subgoup of prime order (in Chinese). Sci. Sin. Math., 40(11), 1055–1078 (2010)
Xu, X. Z.: Saturated fusion systems over a class of finite p-groups. J. Algebra, 413, 135–152 (2014)
Yang, S., Gao, S.: On the control of fusion in the local category for the p-block with a minimal nonabelian defect group. Sci. China. Math., 54, 325–340 (2011)
Zeng, J. W.: An inquality of block invariants. J. Algebra, 193, 724–727 (1997)
Acknowledgements
The authors are grateful to Prof. Jiping Zhang for his helpful suggestions and constructive criticism. Thanks are also given to the referee who made many valuable comments on this paper.
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Supported by National Natural Science Foundation of China (Grant No. 13101193); Zhejiang Natural Science Foundation (Grant No. LY16A010016)
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Yang, S., Gao, S. & Xu, C.B. On the number of irreducible characters in a 3-block with a minimal nonabelian defect group. Acta. Math. Sin.-English Ser. 33, 1267–1274 (2017). https://doi.org/10.1007/s10114-017-5792-4
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DOI: https://doi.org/10.1007/s10114-017-5792-4