Abstract
P. Hall introduced the concept of isoclinism of groups to classify p-groups. It is well-known that two isoclinic nilpotent groups have the same nilpotency class. In this paper using the classification of James of p-groups of order at most p 5 via their isoclinism classes, the degrees of irreducible characters with their frequencies are found. To do this we use the concept of generalized Camina pairs. We also investigate that whether a nonlinear irreducible character can be obtained as a product of two other nonlinear irreducible characters of same degree.
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References
Adan-Bante, E.: Products of characters and finite p-groups. J. Algebra 277(1), 236–255 (2004)
Adan-Bante, E.: Products of characters and finite p-groups. II, Arch. Math. (Basel) 82(4), 289–297 (2004)
Adan-Bante, E.: Restriction of characters and products of characters. Israel J. Math. 174, 221–225 (2009)
Adan-Bante, E., Loukaki, M., Moretó, A.: Homogeneous products of characters. J. Algebra 274(2), 587–593 (2004)
Berkovich, Y.: Groups of prime power order. Vol. 1, 46 of de Gruyter Expositions in Mathematics, With a foreword by Zvonimir Janko. Walter de Gruyter GmbH & Co. KG, Berlin (2008)
Bioch, J.C.: On n-isoclinic Groups. Indag. Math. Proc. 79(5), 400–407 (1976)
Camina, A.R.: Some conditions which almost characterize Frobenius group. Israel J. Math. 31, 153–160 (1978)
Hall, P.: The classification of prime-power groups. J. Reine Angew. Math. 182, 130–141 (1940)
Isaacs, I.M.: Character theory of finite groups. AMS Chelsea Publishing, Academic Press, New York (2000)
James, R.: The groups of order p 6 (p an odd prime). Math. Comp. 34(150), 613–637 (1980)
Lewis, M.L.: The vanishing-off subgroup. J. Algebra 321(4), 1313–1325 (2009)
Lewis, M.L.: Character tables of groups where all nonlinear irreducible characters vanish off the center, In: Proceedings Ischia Group Theory (2008)
Lewis, M.L.: On p-group Camina pairs. J. Group Theory 15, 469–483 (2012)
Manz, O., Wolf, T.R.: Representations of solvable groups, London Mathematical Society Lecture Note Series, 185. Cambridge University Press, Cambridge (1993)
Acknowledgments
The first author is supported by Lady Davis fellowship and thankful to Aner Shalev for his support in this research work. We would like to say thanks to the anonymous referee for his/her useful suggestions and comments, which substantially improved the quality of the paper. The proof of Theorem 1.1 is due to him, which is without using the classification and much easier than our original one. This also inspired us to construct Theorem 1.2 and Proposition 6.1.
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Presented by Radha Kessar.
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Prajapati, S.K., Darafsheh, M.R. & Ghorbani, M. Irreducible Characters of p-group of Order ≤ p 5 . Algebr Represent Theor 20, 1289–1303 (2017). https://doi.org/10.1007/s10468-017-9687-y
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DOI: https://doi.org/10.1007/s10468-017-9687-y