Subgraph of compatible action graph for finite cyclic groups of p-power order

• Main Authors: Shahoodh, Mohammed Khalid, Mohd Sham, Mohamad, Yuhani, Yusof, Sahimel Azwal, Sulaiman, Adam Shariff Adli, Aminuddin
• Format: Conference or Workshop Item
• Language: English; English
• Published: Universiti Malaysia Pahang 2018
• Subjects: LB2300 Higher Education; QA Mathematics
• Online Access: http://umpir.ump.edu.my/id/eprint/24330/1/53.%20Subgraph%20of%20compatible%20action%20graph%20for%20finite.pdf; http://umpir.ump.edu.my/id/eprint/24330/2/53.1%20Subgraph%20of%20compatible%20action%20graph%20for%20finite.pdf; http://umpir.ump.edu.my/id/eprint/24330/
• Institution: Universiti Malaysia Pahang

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.