Cycles and Bipartite Graph on Conjugacy Class of Groups

Abstract

-Let G be a finite non abelian group and B(G) be the bipartite divisor graph of a finite group related to the conjugacy classes of G. We prove that B(G) is a cycle if and only if B(G) is a cycle of length 6 and G A Â SL 2 (q), where A is abelian, and q P f4; 8g. We also prove that if G=Z(G) simple, where Z(G) is the center of G, then B(G) has no cycle of length 4 if and only if G A Â SL 2 (q), where q P f4; 8g.Introduction and results.

Bipartite Divisor Graph for the Product of Subsets of Integers

“…If B(G) is isomorphic to P 4 or P 3 , then σ * (G) = 2. Suppose that G is not solvable; then by [18] we conclude that B(G) is a cycle of length six which is in contradiction to the hypothesis, so G is solvable. Finally, suppose that B(G) is isomorphic to P 2 .…”

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

“…Therefore, it is interesting to see, if one minded so, whether there exists a group G with B(Cl(G)) isomorphic to a cycle. Taeri in [30] has answered this question and has proved that B(Cl(G)) is a cycle if and only if it is the cycle C 6 of length six. Moreover, Taeri has classified the groups G with B(Cl(G)) ∼ = C 6 ; indeed, G ∼ = A × SL 2 (q) where A is an abelian group and q ∈ {4, 8}.…”

Groups having complete bipartite divisor graphs for their conjugacy class sizes

“…For instance, in [9], the first author of this paper and Iranmanesh have classified the groups whose bipartite divisor graphs are paths. Similarly, Taeri [13] considered the case that the bipartite divisor graph is a cycle, and in the course of his investigation posed the following question: Question. ([13, Question 1]) Is there any finite group G such that B(G) is isomorphic to a complete bipartite graph K m,n , for some positive integers m, n ≥ 2?…”

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Given a finite group G, the bipartite divisor graph for its conjugacy class sizes is the bipartite graph with bipartition consisting of the set of conjugacy class sizes of G \ Z(G) (where Z(G) denotes the centre of G) and the set of prime numbers that divide these conjugacy class sizes, and with {p, n} being an edge if gcd(p, n) = 1.In this paper we construct infinitely many groups whose bipartite divisor graph for their conjugacy class sizes is the complete bipartite graph K2,5, giving a solution to a question of Taeri [13].

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“…In this paper we construct infinitely many groups whose bipartite divisor graph for their conjugacy class sizes is the complete bipartite graph K2,5, giving a solution to a question of Taeri [13].…”