Subgraph of Compatible Action Graph for Finite Cyclic Groups of p-Power Order
| Autor: | Yuhani Yusof, Sahimel Azwal Sulaiman, Mohammed Khalid Shahoodh, Mohd Sham Mohamad |
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| Rok vydání: | 2019 |
| Předmět: | |
| Zdroj: | Journal of Physics: Conference Series. 1366:012064 |
| ISSN: | 1742-6596 1742-6588 |
| DOI: | 10.1088/1742-6596/1366/1/012064 |
| Popis: | 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. |
| Databáze: | OpenAIRE |
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