Bipartite divisor graph for the set of irreducible character degrees

Document Type : Research Paper




‎Let $G$ be a finite group‎. ‎We consider the set of the irreducible complex characters of $G$‎, ‎namely $Irr(G)$‎, ‎and the related degree set $cd(G)=\{\chi(1)‎ : ‎\chi\in Irr(G)\}$‎. ‎Let $\rho(G)$ be the set of all primes which divide some character degree of $G$‎. ‎In this paper we introduce the bipartite divisor graph for $cd(G)$ as an undirected bipartite graph with vertex set $\rho(G)\cup (cd(G)\setminus\{1\})$‎, ‎such that an element $p$ of $\rho(G)$ is adjacent to an element $m$ of $cd(G)\setminus\{1\}$ if and only if $p$ divides $m$‎. ‎We denote this graph simply by $B(G)$‎. ‎Then by means of combinatorial properties of this graph‎, ‎we discuss the structure of the group $G$‎. ‎In particular‎, ‎we consider the cases where $B(G)$ is a path or a cycle‎.


Main Subjects

[1] M. Bianchi, D. Chillag, M. L. Lewis and E. Paci ci, Character degree graphs that are complete graphs, Proc. Amer. Math. Soc., 135 (2007) 671-676.
[2] D. Bubb oloni, S. Dol , M. A. Iranmanesh and C. E. Praeger, On bipartite divisor graphs for group conjugacy class sizes, J. Pure Appl. Algebra, 213 (2009) 1722-1734.
[3] S. C. Garrison, On groups with a small number of character degrees, Ph. D. Thesis, University of Wisconsin, Madison, 1973.
[4] M. A. Iranmanesh and C. E. Praeger, Bipartite divisor graphs for integer subsets, Graphs Combin., 26 (2010) 95-105.
[5] I. M. Isaacs, Character theory of nite groups, Academic Press, New York, 1976.
[6] M. L. Lewis, Irreducible character degree sets of solvable groups, J. Algebra, 206 (1998) 208-234.
[7] M. L. Lewis, A Solvable group whose character degree graph has diameter 3, Proc. Amer. Math. Soc., 130 (2001) 625-630.
[8] M. L. Lewis, Solvable groups whose degree graphs have two connected comp onents, J. Group Theory, 4 (2001) 255-275.
[9] M. L. Lewis, Derived lengths of solvable groups having ve irreducible character degrees I, Algeb. Represent. Theory, 4 (2001) 469-489.
[10] M. L. Lewis, An overview of graphs asso ciated with character degrees and conjugacy class sizes in nite groups, Rocky Mountain J. Math., 38 (2008) 175-211.
[11] M. L. Lewis and Q. Meng, Square character degree graphs yield direct pro ducts, J. Algebra, 49 (2012) 185-200.
[12] M. L. Lewis, A. Moreto and T. R. Wolf, Nondivisibility among character degrees, J. Group Theory, 8 (2005) 561-588.
[13] M. Lewis and D. White, Four-vertex degree graphs of nonsolvable groups, J. Algebra, 378 (2013) 1-11.
[14] H. LiGuo and Q. GuoHua, Graphs of nonsolvable groups with four degree-vertices, Sci. China Math. , 58 (2015) 1305-1310.
[15] G. Malle and A. Moreto, Nonsolvable groups with few character degrees, J. Algebra, 294 (2005) 117-126.
[16] S. A. Mo osavi, On bipartite divisor graph for character degrees, Int. J. Group Theory , 6 (2017) 1-7.
[17] T. Noritzsch, Groups having three complex irreducible character degrees, J. Algebra, 175 (1995) 767-798.
[18] A. Previtali, Orbit lengths and character degrees in a p -Sylow subgroup of some classical Lie groups, J. Algebra, 177 (1995) 658-675.
[19] J. M. Riedl, Fitting heights of o dd-order groups with few character degrees, J. Algebra, 267 (2003) 421-442.
[20] B. Taeri, Cycles and bipartite graph on conjugacy class of groups, Rend. Sem. Mat. Univ. Padova., 123 (2010) 233-247.
[21] H. P. Tong-Viet, Groups whose prime graph has no triangles, J. Algebra, 378 (2013) 196-206.
Volume 6, Issue 4 - Serial Number 4
December 2017
Pages 41-51
  • Receive Date: 01 July 2016
  • Revise Date: 26 January 2017
  • Accept Date: 04 February 2017
  • Published Online: 01 December 2017