A SURVEY ON THE AUTOMORPHISM GROUPS OF THE COMMUTING GRAPHS AND POWER GRAPHS

Let G be a nite group. The power graph P(G) of a group G is the graphwhose vertex set is the group elements and two elements are adjacent if one is a power of the other. The commuting graph \Delta(G) of a group G, is the graph whose vertices are the group elements, two of them joined if they commute. When the vertex set is G-Z(G), this graph is denoted by \Gamma(G). Since the results based on the automorphism group of these kinds of graphs are so sporadic, in this paper, we give a survey of all results on the automorphism group of power graphs and commuting graphs obtained in the literature.

The Probability That an Ordered Pair of Elements is an Engel Pair

Let G be a nite group. We denote by ep(G) the probability that[x;n y] = 1 for two randomly chosen elements x and y of G and some posi-tive integer n. For x 2 G we denote by EG(x) the subset fy 2 G : [y;n x] =1 for some integer ng. G is called an E-group if EG(x) is a subgroup of G for allx 2 G. Among other results, we prove that if G is an non-abelian E-group withep(G) 16 , then G is not simple and minimal non-solvable.