On regular bipartite divisor graph for the set of irreducible character degrees

Year 2019, Volume: 48 Issue: 6 , 1620 - 1625 , 08.12.2019

Roghayeh Hafezieh

https://izlik.org/JA76PW53EX

Abstract

Given a finite group $G$, the \textit{bipartite divisor graph}, denoted by $B(G)$, for its irreducible character degrees is the bipartite graph with bipartition consisting of $cd(G)^{*}$, where $cd(G)^{*}$ denotes the nonidentity irreducible character degrees of $G$ and the $\rho(G)$ which is the set of prime numbers that divide these degrees, and with $\{p,n\}$ being an edge if $\gcd(p,n)\neq 1$. In [Bipartite divisor graph for the set of irreducible character degress, Int. J. Group Theory, 2017], the author considered the cases where $B(G)$ is a path or a cycle and discussed some properties of $G$. In particular she proved that $B(G)$ is a cycle if and only if $G$ is solvable and $B(G)$ is either a cycle of length four or six. Inspired by $2$-regularity of cycles, in this paper we consider the case where $B(G)$ is an $n$-regular graph for $n\in\{1,2,3\}$. In particular we prove that there is no solvable group whose bipartite divisor graph is $C_{4}+C_{6}$.

Keywords

bipartite divisor graph , irreducible character degrees , regular graph

References

• [1] R. Hafezieh, Bipartite divisor graph for the set of irreducible character degress, Int. J. Group Theory 6 (4), 41-51, 2017.
• [2] B. Huppert and W. Lempken, Simple groups of order divisible by at most four primes, Proceeding of F. Scorina Gemel State University 16 (3), 64-75, 2000.
• [3] M.A. Iranmanesh and C.E. Praeger, Bipartite divisor graphs for integer subsets, Graphs Combin. 26, (2010), 95-105.
• [4] I.M. Isaacs, Character theory of finite groups, Academic Press, New York, 1976.
• [5] D.M. Kasyoki, Finite Solvable Groups with 4-Regular Prime Graphs, African Institute for Mathematical Sciences, Master Thesis, 2013.
• [6] M.L. Lewis, Determining group structure from sets of irreducible character degrees, J. Algebra 206, 235-260, 1998.
• [7] M.L. Lewis, Solvable groups whose degree graphs have two connected components, J. Group Theory 4, 255-275, 2001.
• [8] M.L. Lewis, An overview of graphs associated with character degrees and conjugacy class sizes in finite groups, Rocky Mountain J. Math. 38, 175-211, 2008.
• [9] M.L. Lewis and Q. Meng, Square character degree graphs yield direct products, J. Algebra 349, 185-200, 2012.
• [10] M.L. Lewis and D. L. White, Four-vertex degree graphs of nonsolvable groups, J. Algebra 378, 1-11, 2013.
• [11] T. Noritzsch, Groups having three complex irreducible character degrees, J. Algebra, 175, 767-798, 1995.
• [12] H.P. Tong-Viet, Groups whose prime graphs have no triangles, J. Algebra 378, 196- 206, 2013.
• [13] H.P. Tong-Viet, Finite groups whose prime graphs are regular, J. Algebra 397, 18-31, 2014.