Subgraph of compatible action graph for finite cyclic groups of p-power order - Shahoodh, Mohammed Khalid; Mohd Sham, Mohamad; Yuhani, Yusof; Sahimel Azwal, Sulaiman; Adam Shariff Adli, Aminuddin (Universiti Malaysia Pahang, 2018)
Given two groups G and H, then the nonabelian tensor product GH is the group generated by gh satisfying the relations ()()gggghghgh′′⊗=⊗⊗and()()hhghhghgh′′ for all ,ggG′∈ and ,hhH′∈. If G and H act on each other and each of which acts on itself by conjugation and satisfying 1() (())ghghggg−′′= and 1() (())hghghhh−′′=, then the actions are said to be compatible. The action of G on H, gh is a homomorphism from G to a group of automorphism H. If (,g)ghh be a pair of the compatible actions for the nonabelian tensor product of GH, then GHGH GHEΓ is a compatible action graph with the set of vertices, ()GHVΓ and the set of edges, ()GHEΓ. In this paper, the necessary and sufficient conditions for the cyclic subgroups ofp-power order acting on each other in a compatible way are given. Hence, a subgraph of a compatible action graph is introduced and its properties are given.