Products of conjugacy classes and irreducible characters in finite camina groups

Articles in Press

Document Type : Research Paper

Authors

School of Mathematical Science, Tianjin Normal University, Tianjin 300387, P. R. China

Abstract

This paper investigates two problems concerning products of conjugacy classes and irreducible characters in finite groups. First, we provide a simplified proof of Dade and Yadav's theorem concerning groups in which the conjugacy class product $x^Gy^G=(xy)^G$ holds for all $x\in G, y\in G-(x^{-1})^G$. By using the classification of Camina groups due to Dark and Scoppola, we demonstrate that such groups are either abelian, non-abelian Camina $p$-groups, or specific Frobenius groups of the form $F^{+}\rtimes F^{\times}$ for some finite field $F$ with $|F|>2$ or $E_9\rtimes Q_8$. Second, we explore the dual problem for characters: groups in which the product of any two non-conjugate irreducible characters has exactly one irreducible constituent. We establish that such groups are precisely the Camina $p$-groups and the two families of Frobenius groups mentioned above.

Keywords

Main Subjects

References

[1] E. Adan-Bante, M. Loukaki and A. Moret´o, Homogeneous products of characters, J. Algebra, 274 no. 2 (2004) 587–593.
[2] Z. Arad and M. Herzog, Products of conjugacy classes in groups, Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1985.
[3] C. Bessenrodt and A. S. Kleshchev, On Kronecker products of complex representations of the symmetric and alternating groups, Pac. J. Math., 190 (1999) 201–223.
[4] C. Bessenrodt and A. S. Kleshchev, Irreducible tensor products over alternating groups, J. Algebra, 228 (2000) 536–550.
[5] A. R. Camina, Some conditions which almost characterize Frobenius groups, Isr. J. Math., 31 (1978) 153–160.
[6] E. C. Dade and M. K. Yadav, Finite groups with many product conjugacy classes, Isr. J. Math., 154 (2006) 29–49.
[7] R. Dark and C. M. Scoppola, On Camina group of prime power order, J. Algebra, 181 (1996) 787–802.
[8] H. Fukushima, Irreducible products of characters of solvable groups, J. Algebra, 321 no. 1 (2009) 312–315.
[9] R. Guralnick, G. Malle and P. H. Tiep, Products of conjugacy classes in finite and algebraic simple groups, Adv. Math., 234 (2013) 618–652.
[10] I. M. Isaacs, Irreducible products of characters, J. Algebra, 223 no. 2 (2000) 630–646.
[11] I. M. Isaacs, Character theory of finite groups, AMS Chelsea Publishing, Providence, RI, 2006.
[12] I. M. Isaacs and M. L. Lewis, Camina p-groups that are generalized Frobenius complements, Arch. Math. (Basel), 104 (2015) 401–405.
[13] P. Jin and Y. Yang, On irreducible products of characters, Math. Nachr., 294 no. 11 (2021) 2184–2187.
[14] M. Larsen and P. H. Tiep, Uniform character bounds for finite classical groups, Annals of Math., 200 (2024) 1–70.
[15] M. Larsen and P. H. Tiep, Character estimates for finite classical groups and the asymptotic Thompson conjecture, Preprint, arXiv:2403.09047.
[16] M. L. Lewis, Camina groups, Camina pairs, and generalizations, Group Theory and Computation, 141–173, Indian Stat. Inst. Ser., Springer, Singapore, 2018.
[17] M. L. Lewis, Classifying Camina groups: a theorem of Dark and Scoppola, Rocky Mountain J. Math., 44 (2014) 591–597. See also: Erratum on Classifying Camina groups: a theorem of Dark and Scoppola, Rocky Mountain J. Math., 45 (2015) 273.
[18] K. Magaard and P. H. Tiep, Irreducible tensor products of representations of finite quasi-simple groups of Lie type, Modular Representation Theory of Finite Groups, 239–262, Walter de Gruyter, Berlin, 2021.
[19] G. Navarro and P. H. Tiep, On irreducible products of characters, J. Algebra, 573 (2021) 38–55.
[20] I. Zisser, Irreducible products of characters in An, Isr. J. Math., 84 (1993) 147–151.
International Journal of Group Theory

Articles in Press, Corrected Proof
Available Online from 15 July 2026

Files

History

  • Receive Date: 07 February 2026
  • Revise Date: 23 June 2026
  • Accept Date: 03 July 2026
  • Published Online: 15 July 2026

Share

How to cite

Statistics