Automorphism groups of the power graph of a finite group

For a finite group G, the power graph P(G) is a graph with the vertex set G, in which two distinct elements are adjacent if one is a power of the other. Th e study of power graph was started by publishing the seminal paper of Kelarev and Quinn [7]. In this paper, the authors considered the directed power graph of groups and semigroups into account. Th e main result of the mentioned paper gives a very technical description of the power graph structure of finite abelian groups. Th e same authors [8] studied also the power graph of the multiplicative subsemigroup of the ring of n × n matrices over a skew-field. Th e interested readers can be consulted [5, 6] for more information about the power graphs of semigroups. Chakrabarty et al. [4] introduced the undirected power graph of a finite group and proved that this graph is complete if and only if G is a cyclic p􀀀group, for a prime number p. Cameron and Ghosh [2] proved that two abelian groups with isomorphic power graphs must be isomorphic and conjectured that two finite groups with isomorphic power graphs have the same number of elements of each order. Cameron [3] responded affirmatively this conjecture. We refer the interested readers to [9, 10, 11], for more information on the power graph and the automorphism group of certain finite groups. In this talk, we will report recent results on automorphism groups of P(G). As an important result, the automorphism group of this graph can be written as a combination of Cartesian and wreath products of some symmetric groups. Th e full automorphism groups of this graph of certain finite groups is also calculated.