On a Graph Related to Conjugacy Classes of Groups

Abstract

Let G be a finite group. Attach to G the following graph Γ: its vertices are the non‐central conjugacy classes of G, and two vertices are connected if their cardinalities are not coprime. Denote by n(Γ) the number of the connecte components of Γ. We prove that n(Γ) ⩽ 2 for all finite groups, and we completely characterize groups with n(Γ) = 2. When Γ is connected, then the diameter of the graph is at most 4. For simple non‐abelian finite groups, the graph is complete. Similar results are proved for infinite FC‐groups.

Relative non nil-n graphs of finite groups

“…It is clear that if n = 1 then P (1) nil (H, G) = d(H, G) which is the relative commutativity degree of a subgroup H in the group G (see Ref. 10) and when H = G then P (1) nil (G) = d(G) is the commutativity degree of the group G (see Refs.…”

“…10) and when H = G then P (1) nil (G) = d(G) is the commutativity degree of the group G (see Refs. 11, 12).…”

Coprime Conjugacy Class Sizes

“…To prove this theorem we start with a lemma, which relies on the concept of a kernel of a subset A of a group G: ker(A) := {x ∈ G : xA = A} [9]. The kernel is a subgroup of G and since A is a union of cosets of ker(A) it follows that the order of ker(A) divides the order of A.…”

Inverse graphs associated with finite groups

“…For an excellent survey on the Hamiltonian cycles in Cayley graphs see [17]. Aside this, several other graphs have also been associated with finite groups such as commuting graphs, intersection graphs, prime graphs, non-commuting graphs, conjugacy class graphs, etc, [5,6,11,12,18]. In order to justify our claim that the inverse graph is new, we as well illustrate by examples how it is different from some known graphs associated with groups.…”