Groups having complete bipartite divisor graphs for their conjugacy class sizes

Abstract

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].

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

“…Here we observe that there exist groups G where B(G) is an arbitrary complete bipartite graph. The analogous problem for the bipartite divisor graph for the set of conjugacy class sizes seems considerably harder; it is widely open and it is stated in [13]. Let k be a positive integer and let G := H k be the Cartesian product of k copies of H. Clearly, cd(G) = {1, n, n 2 , .…”

“…Since C 4 is also the complete bipartite graph K 2,2 and since there is no finite group G with B(Cl(G)) ∼ = K 2,2 , Taeri [30, Question 1] has asked whether B(Cl(G)) can be isomorphic to some complete bipartite graph. In [13], we answered this question and we constructed infinitely many groups G with B(Cl(G)) ∼ = K 2,5 . However, as far as we are aware, it is not known for which positive integers n and m there exists a finite group G with B(Cl(G)) ∼ = K n,m (let alone a meaningful classification of the groups G with B(Cl(G)) ∼ = K n,m ).…”

“…13 • 3 • 11; moreover, G contains a Hall 2 ′ -subgroup K with K cyclic and G contains a normal 2-subgroup Q with |G : QK| = 2. Furthermore, Q/γ 2 (Q) is elementary abelian of order 2 10 = 1 024, γ 2 (Q) is elementary abelian of order 2 2 = 4 and K centralizes γ 2 (Q).…”