Abstract
Let G be a finite nonabelian group and Kern(G) the set of kernels of nonlinear irreducible characters of G. In this paper, finite p-groups G with ∣Kern(G)∣ ≤ 3 are determined.
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References
Berkovich Y., Groups of Prime Power Order, Vol. 1. Berlin: Walter de Gruyter, 2008
Berkovich Y.G., Zhmud’ E.M., Characters of Finite Groups Part 2. Providence: American Mathematical Society, 1999
Chen X.Y., Lewis M., Groups with only two nonlinear non-faithful irreducible characters. Czechoslovak Math. J., 2019, 69(2): 427–429
Fang X.G., An L.J., A classification of finite metahamiltonian p-groups. Commun. Math. Stat., 2021, 9(2): 239–260
Fernández-Alcober G.A., Moretò A., Groups with two extreme character degrees and their normal subgroups. Trans. Amer. Math. Soc., 2001, 353(6): 2171–2192
Isaacs I.M., Character Theory of Finite Groups. New York: Academic Press, 1976
Iranmanesh A., Saeidi A., Finite groups with a unique nonlinear nonfaithful irreducible character. Arch. Math. (Brno), 2011, 47(2): 91–98
Li Y.L., Finite solvable groups with exactly two nonlinear non-faithful irreducible characters. J. Algebra Appl., 2019, 18(5): 1950091, 7 pp.
Li Y.L., Chen X.Y., Li H.M., Finite p-groups with exactly two nonlinear non-faithful irreducible characters. Czechoslovak Math. J., 2019, 69(1): 173–181
Qian G.H., Wang Y.M., A note on character kernels in finite groups of prime power order. Arch. Math. (Basel), 2008, 90(3): 193–199
Saeidi A., Classification of solvable groups possessing a unique nonlinear non-faithful irreducible character. Cent. Eur. J. Math., 2014, 12(1): 79–83
Seitz G., Finite groups having only one irreducible representation of degree greater than one. Proc. Amer. Math. Soc., 1968, 19: 459–461
Xu M.Y., An L.J., Zhang Q.H., Finite p-groups all of whose non-abelian proper subgroups are generated by two elements. J. Algebra, 2008, 319: 3603–3620
Zhang G.X., Finite groups with exactly two nonlinear irreducible characters. Chinese Ann. Math. Ser. A, 1996, 17(2): 227–232 (in Chinese)
Zhang Q.H., Zhao L.B., Li M.M., Shen Y.Q., Finite p-groups all of whose subgroups of index p3 are abelian. Commun. Math. Stat., 2015, 3: 69–162
Acknowledgements
The authors would like to thank the reviewers for their valuable suggestions and comments on this paper. Thanks also to Professor Guohua Qian for his kind reading and suggestion before the paper being submitted. This work was supported by the National Natural Science Foundation of China (Nos. 11771258, 11971280, 11801341).
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Li, P., Zhang, Q. Finite p-groups with Few Kernels of Nonlinear Irreducible Characters. Front. Math 18, 65–80 (2023). https://doi.org/10.1007/s11464-021-0307-0
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DOI: https://doi.org/10.1007/s11464-021-0307-0