$p$-groups with a small number of character degrees and their normal subgroups

Document Type : Research Paper

Authors

Department of Mathematics, Cotton University, Guwahati, India

Abstract

If $G$ be a finite $p$-group and $\chi$ is a non-linear irreducible character of $G$, then $\chi(1)\leq |G/Z(G)|^{\frac{1}{2}}$. In \cite{fernandez2001groups}, Fern'{a}ndez-Alcober and Moret'{o} obtained the relation between the character degree set of a finite $p$-group $G$ and its normal subgroups depending on whether $|G/Z(G)|$ is a square or not. In this paper we investigate the finite $p$-group $G$ where for any normal subgroup $N$ of $G$ with $G'\not \leq N$ either $N\leq Z(G)$ or $|NZ(G)/Z(G)|\leq p$ and obtain some alternate characterizations of such groups. We find that if $G$ is a finite $p$-group with $|G/Z(G)|=p^{2n+1}$ and $G$ satisfies the condition that for any normal subgroup $N$ of $G$ either $G'\not \leq N$ or $N\leq Z(G)$, then $cd(G)=\{1, p^{n}\}$. We also find that if $G$ is a finite $p$-group with nilpotency class not equal to $3$ and $|G/Z(G)|=p^{2n}$ and $G$ satisfies the condition that for any normal subgroup $N$ of $G$ either $G'\not \leq N$ or $|NZ(G)/Z(G)|\leq p$, then $cd(G) \subseteq \{1, p^{n-1}, p^{n}\}$.

Keywords

Main Subjects

References

[1] H. Doostie and A. Saeidi, Finite p-groups with few non-linear irreducible character kernels, Bull. Iranian Math. Soc., 38 no. 2 (2012) 413–422.
[2] G. Fernández-Alcober and A. Moretó, Groups with two extreme character degrees and their normal subgroups, Trans. Amer. Math. Soc., 353 no. 6 (2001) 2171–2192.
[3] M. L. Lewis, The vanishing-off subgroup, J. Algebra, 321 no. 4 (2009) 1313–1325.
[4] S. K. Prajapati, M. R. Darafsheh and M. Ghorbani, Irreducible characters of p-group of order ≤ p5 , Algebr. Represent. Theory, 20 no. 5 (2017) 1289–1303.
[5] G. Qian and W. Yanming, A note on character kernels in finite groups of prime power order, Arch. Math. (Basel), 90 no. 3 (2008) 193–199.
[6] B. Huppert, Character theory of finite groups, De Gruyter Expositions in Mathematics, 25, Walter de Gruyter & Co., Berlin, 1998.
[7] I. M. Isaacs, Character theory of finite groups, Dover Publications, Inc., New York, 1994.
[8] I. M. Isaacs, Finite Group Theory, American Mathematical Society, Rhode Island, 2008.
[9] H. Kurzweil and B. Stellmacher, The theory of finite groups. An introduction, Translated from the 1998 German original, Universitext, Springer-Verlag, New York, 2004.
[10] J. J. Rotman, An introduction to the theory of groups, Fourth edition, Graduate Texts in Mathematics, 148, Springer-Verlag, New York, 1995.
International Journal of Group Theory
Volume 14, Issue 3 - Serial Number 3
September 2025
Pages 171-180

Files

History

  • Receive Date: 22 March 2024
  • Revise Date: 23 September 2024
  • Accept Date: 23 September 2024
  • Published Online: 12 October 2024

Share

How to cite

Statistics