On Commuting Graphs for Elements of Order 3 in Symmetric Groups

Abstract

The commuting graph $\mathcal{C}(G,X)$, where $G$ is a group and $X$ is a subset of $G$, is the graph with vertex set $X$ and distinct vertices being joined by an edge whenever they commute. Here the diameter of $\mathcal{C}(G,X)$ is studied when $G$ is a symmetric group and $X$ a conjugacy class of elements of order $3$.

Commuting Graphs, C(G, X) in Symmetric Groups Sym(n) and Its Connectivity

“…By using Theorems 1 and 2 where appropriate, we can now put these to good use to partially prove the following result (note that this observation is based on the results in Table 1 of [2] and Table 1 of [3]): Theorem 3. Let G = Sym(n) and t ∈ G be of cycle type 3 r with r ≥ 1.…”

“…, C 10 are the connected components of C(G, X). Then, (1,4,6) (2,3,5), (1,6,4)(2, 3, 5)} C 9 = {(1, 5, 3)(2, 4, 6), (1,3,5) (2,4,6), (1,5,3) (2,6,4), (1,3,5)(2, 6, 4)} C 10 = {(1, 4, 3)(2, 6, 5), (1,3,4) (2,6,5), (1,4,3)(2, 5, 6), (1,3,4)(2, 5, 6)} Therefore, when n = 6 and r = 2, C(G, X) consists of ten connected components each of size four. Proposition 2.…”

“…It shows an absolute contrast to the case having G as the symmetric group of degree 6, Sym(6) and X = (1, 2, 3) G , where C(G, X) is disconnected even though both cases deal with the set of elements of cycle type 3 1 as the vertex set. More specific examples of connected and disconnected commuting graphs can be referred from the results in Table 1 of [2] and Table 1 of [3]. Evidently, there are certain cases where C(G, X) is not connected or disconnected, which further motivate us to categorize them.…”

“…5)} C 2 = {(1,2,4),(1,4,2),(3,5,6),(3,6,5)} C 3 = {(1, 2, 5),(1,5,2),(3,4,6),(3,6,4)} C 4 = {(1, 2, 6), (1, 6, 2), (3, 4, 5), (3, 5, 4)} C 5 = {(1, 3, 4), (1, 4, 3), (2, 5, 6), (2, 6, 5)} C 6 = {(1, 3, 5), (1, 5, 3), (2, 4, 6), (2, 6, 4)} C 7 = {(1, 3, 6), (1, 6, 3), (2, 4, 5), (2, 5, 4)} C 8 = {(1, 4, 5), (1, 5, 4), (2, 3, 6), (2, 6, 3)} C 9 = {(1, 4, 6), (1, 6, 4), (2, 3, 5), (2, 5, 3)} C 10 = {(1, 5, 6), (1, 6, 5)(2, 3, 4), (2, 4, 3)}Therefore, when n = 6 and r = 1, C(G, X) consists of ten connected components each of size four. Let G = Sym(6) and t = (1, 2, 3) ∈ G. Then, every connected component of disconnected C(G, X) is isomorphic to the complete graph of order 4, K 4 , as shown inFigure 1.…”

Commuting involution graphs for

“…Segev and Seitz looked in [10] at commuting graphs for finite simple groups G where X consists of the non-identity elements of G. More recently Giudici and Pope [6] gave some results on bounding the diameters of commuting graphs of finite groups. Commuting graphs for elements of order 3 have been considered in [2]; there have also been many papers dealing with the case where X consists of all non-identity elements of a given group such as for example [1].…”