Subgraph of Compatible Action Graph for Finite Cyclic Groups of p-Power Order

Given two groups G and H, then the nonabelian tensor product G ⊗ H is the group generated by g ⨂ h satisfying the relations gg′ ⊗ h = (gg′ ⊗ gh) (g ⨂ h) and g ⊗ hh′ = (g ⊗ h) (hg′ ⊗ hh) for all g g′ ∈ G and h, h′ ∈ H. If G and H act on each other and each of which acts on itself by conjugation and satisfying (g h)g′ = g(h(g-1 g′)) and (h g)h′ = h(g(h-1 h′)), 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 (gh, hg) be a pair of the compatible actions for the nonabelian tensor product of G ⊗ H then ΓG ⊗ H = (V(ΓG ⊗ H), (E(ΓG ⊗ H)) is a compatible action graph with the set of vertices, (V(ΓG ⊗ H) and the set of edges, (E(ΓG ⊗ H). In this paper, the necessary and sufficient conditions for the cyclic subgroups of p-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.