On connected components and perfect codes of proper order graphs of finite groups
Let G be a finite group with the identity element e. The proper order graph of G, denoted by â* (G), is an undirected graph with a vertex set G \ {e}, where two distinct vertices x and y are adjacent whenever o(x) | o(y) or o(y) | o(x), where o(x) and o(y) are the orders of x and y, respectively. Th...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
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Estonian Academy Publishers
2024-02-01
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| Series: | Proceedings of the Estonian Academy of Sciences |
| Subjects: | |
| Online Access: | https://kirj.ee/wp-content/plugins/kirj/pub/proc-1-2024-68-76_20240202102007.pdf |
| Summary: | Let G be a finite group with the identity element e. The proper order graph of G, denoted by â* (G), is an undirected graph with a vertex set G \ {e}, where two distinct vertices x and y are adjacent whenever o(x) | o(y) or o(y) | o(x), where o(x) and o(y) are the orders of x and y, respectively. This paper studies the perfect codes of â*(G). We characterize all connected components of a proper order graph and give a necessary and sufficient condition for a connected proper order graph. We also determine the perfect codes of the proper order graphs of a few classes of finite groups, including nilpotent groups, CP-groups, dihedral groups and generalized quaternion groups. |
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| ISSN: | 1736-6046 1736-7530 |