Automorphism Group of Power Graphs of Finite Groups

The directed power graph of a semigroup S was defined by Kelarev and Quinn as the digraph ( ) S with vertex set S , in which there is an arc from x to y if and only if m x y  or m y x  for some positive integer m. Motivated by this, Chakrabarty et al. defined the (undirected) power graph ( ) S , in which distinct x and y are joined if one is a power of the other. The concept of power graphs has been studied extensively by many authors. Let  be a graph. We denote V ( )  and E ( )  for vertices and edges of  , respectively. The (open) neighborhood N a( ) of vertex a V ( ) is the set of vertices adjacent to a. Also the closed neighborhood of a , N a N a a [ ] ( ) { }  . Throughout this paper, all groups and graphs are assumed to be finite and the following notation is used: Aut G( ) denotes the group of automorphisms of G ; Z m the cyclic group of order m ; n Z m the direct product of n copies of Z m .

The power graph of a group is the graph whose vertex set is the set of nontrivial elements of the group and two elements are adjacent if one is a power of the other. We introduce some ways to find the automorphism groups of some graphs. Let L be a graph and x y V L N y N x    { ( ) | [ ] [ ]}. We define the weighted graph L as follows: V L x x V L ( ) { | ( )}   , weight x x ( ) | |  , And two vertices x and y are adjacent if x and y are adjacent in L . As an application, we describe the automorphism group of the power graph of a finite group G as: | | ( ( )) ( ( )) ( ( )) . x x V P G Aut P G Aut P G S    Let G be a finite nilpotent group, 1 2 1 2 | | t n n n G p p p  t and G P P P     1 2 t where | | i n P p i i  .We obtain the automorphism group of the power graph of abelian and nilpotent groups by their sylow subgroups which is: 1 2 | | ( ( )) ( ( )) ( ( ( )) ( ( )) ( ( ))) . t x x V P G Aut P G Aut P P Aut P P Aut P P S       Finally, we calculate the automorphism group of the power graph of homocyclic group ( ) m n p G Z  , n  1 as: 1 2 1 1 ( ) ( ( )) (( ( ) ) ) ( ( ) ),