On permutability graphs of subgroups of groups

Source abstract

The permutability graph of subgroups of a given group G, denoted by Γ(G), is a graph with vertex set consists of all the proper subgroups of G and two distinct vertices in Γ(G) are adjacent if and only if the corresponding subgroups permute in G. In this paper, we classify the finite groups whose permutability graphs of subgroups are one of bipartite, star graph, C 3 -free, C 5 -free, K 4 -free, K 5 -free, K 1,4 -free, K 2,3 -free or P n -free (n = 2, 3, 4). We investigate the same for infinite groups also. Moreover, some results on the girth, completeness and regularity of the permutability graphs of subgroups of groups are obtained. Among the other results, we characterize groups Q 8 , S 3 and A 4 by using their permutability graphs of subgroups.