Subgraph of Compatible Action Graph for Finite Cyclic Groups of p-Power Order - Shahoodh, Mohammed Khalid; Mohd Sham, Mohamad; Yuhani, Yusof; Sahimel Azwal, Sulaiman (IOP Publishing, 2019)
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 subgra