Subgraph of compatible action graph for finite cyclic groups of p-power order
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...
| Main Authors: | , , , , |
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| Format: | Conference or Workshop Item |
| Language: | English English |
| Published: |
Universiti Malaysia Pahang
2018
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| Subjects: | |
| Online Access: | http://umpir.ump.edu.my/id/eprint/24330/ 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 |
| Summary: | 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. |
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