Permutability graph of cyclic subgroups

2015/04/03 by R. Rajkumar, Rajkumar, R., P. Devi +1
Mathematics · #05C10 #05C25 #20F16 #FOS: Mathematics #Group Theory (math.GR) #math.GR #msc:05C10 #msc:05C25 #msc:20F16

paper · pdf · doi:10.48550/arxiv.1504.00801

14 pages

arxiv created 2015/04/03 · arxiv updated 2015/04/06

Abstract

Let G be a group. The permutability graph of cyclic subgroups of G, denoted by Γ_c(G), is a graph with all the proper cyclic subgroups of G as its vertices and two distinct vertices in Γ_c(G) are adjacent if and only if the corresponding subgroups permute in G. In this paper, we classify the finite groups whose permutability graph of cyclic subgroups belongs to one of the following: bipartite, tree, star graph, triangle-free, complete bipartite, P_n, C_n, K₄, K_1,3-free, unicyclic. We classify abelian groups whose permutability graph of cyclic subgroups are planar. Also we investigate the connectedness, diameter, girth, totally disconnectedness, completeness and regularity of these graphs.

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