Abstract
Let \(G\) be a finite group. The character degree graph of \(G\), which is denoted by \(\Gamma (G)\), is the graph whose vertices are the prime divisors of the character degrees of the group \(G\) and two vertices \(p_1\) and \(p_2\) are joined by an edge if \(p_1p_2\) divides some character degree of \(G\). In this paper we prove that the simple group \(\mathrm{PSL}(2,p^2) \) is uniquely determined by its character degree graph and its order. Let \(X_1(G)\) be the set of all irreducible complex character degrees of \(G\) counting multiplicities. As a consequence of our results we prove that if \(G\) is a finite group such that \(X_1(G)=X_1(\mathrm{PSL}(2,p^2) )\), then \(G\cong \mathrm{PSL}(2,p^2) \). This implies that \(\mathrm{PSL}(2,p^2) \) is uniquely determined by the structure of its complex group algebra.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Conway, J.H., Curtis, R.T., Norton, S.P., Parker, R.A., Wilson, R.A.: Atlas of finite groups. Oxford University Press, Oxford (1985)
Crescenzo, P.: A diophantine equation which arises in the theory of finite groups. Adv. Math. 17, 25–29 (1975)
Huppert, B.: Character Theory of Finite Groups. De Gruyter, Berlin (1998)
Isaacs, I.M.: Character Theory of Finite Groups. Academic Press, New York (1976)
Isaacs, I.M.: Character degree graphs and normal subgroups. Trans. Amer. Math. Soc. 356, 1155–1183 (2004)
Isaacs, I.M.: Finite Group Theory. Graduate Studies in Mathematics, 92. American Mathematical Society, Providence, RI (2008)
Jiang, Q., Shao, C.: Recognition of \({L_{2}}\)(q) by its group order and largest irreducible character degree. Monatsh. Math. (2014). doi:10.1007/s00605-014-0607-5
Khosravi, B., Khosravi, B., Khosravi, B.: Recognition of PSL(2,\(p\)) by order and some information on its character degrees where \(p\) is a prime. Monatsh. Math. (2014). doi:10.1007/s00605-013-0582-2
Khosravi, B., Khosravi, B., Khosravi, B.: Some extensions of PSL(\(2,p^{2}\)) are uniquely determined by their complex group algebras. Comm. Algebra. to appear
Khosravi, B., Khosravi, B., Khosravi, B., Momen, Z.: A new characterization for the simple group PSL(\(2, p^{2}\)) by order and some character degrees. Czech. Math. J. to appear
Khosravi, B., Khosravi, B., Khosravi, B., Momen, Z.: Recognition by character degree graph and order of the simple groups of order less than 6000. Miskolc Math. Notes. to appear
Khosravi, B., Khosravi, B., Khosravi, B.: A new characterization for some extensions of PSL(2,\(p\)) by order and some character degrees. Submitted
Kurzweil, H., Stellmacher, B.: The theory of finite groups: an introduction. Springer, New York (2004)
Lewis, M.L.: An overview of graphs associated with character degrees and conjugacy class sizes in finite groups. Rocky Mt. J. Math. 38, 175–211 (2008)
Lewis, M.L., White, D.L.: Nonsolvable groups with no prime dividing three character degrees. J. Algebra. 336, 158–183 (2011)
Manz, O., Staszewski, R., Willems, W.: On the number of components of a graph related to character degrees. Proc. Amer. Math. Soc. 103, 31–37 (1988)
Tong-Viet, H.P.: Simple classical groups of Lie type are determined by their character degrees. J. Algebra. 357, 61–68 (2012)
White, D.L.: Degree graphs of simple groups. Rocky Mt. J. Math. 39, 1713–1739 (2009)
White, D.L.: Character degrees of extensions of \(\text{ PSL }_{2}\)(q) and \(\text{ SL }_{2}\)(q). J. Group Theory. 16, 1–33 (2013)
Xu, H., Chen, G.Y., Yan, Y.: A new characterization of simple \(K_{3}\)-groups by their orders and large degrees of their irreducible characters. Comm. Algebra. 42, 5374–5380 (2014)
Xu, H., Yan, Y., Chen, G.Y.: A new characterization of Mathieu-groups by the order and one irreducible character degree. J. Ineq. App 209, 1–6 (2013)
Acknowledgments
The authors would like to thank the referee for his/her valuable suggestions. Also the authors express their gratitude to Professor Geoff Robinson for his helpful guidance. The first author would like to thank the Institute for Research in Fundamental Sciences (IPM) for the financial support. This paper is dedicated to our parents for their unending love and support.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by A. Constantin.
This research was in part supported by a Grant from IPM (No 92050120).
Rights and permissions
About this article
Cite this article
Khosravi, B., Khosravi, B., Khosravi, B. et al. Recognition of the simple group PSL(\(2,p^{2}\)) by character degree graph and order. Monatsh Math 178, 251–257 (2015). https://doi.org/10.1007/s00605-014-0678-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00605-014-0678-3