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′ ⊗ g h) (g ⨂ h) and g ⊗ hh′ = (g ⊗ h) (h g′ ⊗ h h) 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, g h is a homomorphism from G to a group of automorphism H. If (g h, hg) be a pair of the compatible actions for the nonabelian tensor product of G ⊗ H then Γ G ⊗ H = (V(Γ G ⊗ H ), (E(Γ G ⊗ H )) is a compatible action graph with the set of vertices, (V(Γ G ⊗ H ) and the set of edges, (E(Γ G ⊗ 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.

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The compatible actions played an important role before determining the nonabelian tensor product of groups. Different compatible pair of actions gives a different nonabelian tensor product even for the same group. The aim of this paper is to determine the exact number of the compatible pair of actions for the finite cyclic groups of p-power order where p is an odd prime. By using the necessary and sufficient number theoretical conditions for a pair of the actions to be compatible with the actions that have p-power order, the exact number of the compatible pair of actions for the finite cyclic groups of p-power order has been determined and given as a main result in this paper.

Compatible actions are important in determining the non-abelian tensor product. Different compatible pair of actions gives a different tensor product even for the same group. The purpose of this paper is to determine the exact number of compatible pair of actions for the finite cyclic groups of 3-power order. By using some properties of number theory, the number of the compatible pair of actions for finite cyclic groups of 3-power order with a specific order of actions is determined and given as a main result in this paper.

In this paper, the compatible action graph for the finite cyclic groups of p-power order has been considered. The purpose of this study is to introduce some properties of the compatible action graph for finite p-groups.