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A combinatorial property and power graphs of groups. (English) Zbl 0966.05040
Dorninger, D. (ed.) et al., Contributions to general algebra 12. Proceedings of the 58th workshop on general algebra “58. Arbeitstagung Allgemeine Algebra”, Vienna, Austria, June 3-6, 1999. Klagenfurt: Verlag Johannes Heyn. 229-235 (2000).
Authors’ abstract: The power graph of a group $$G$$ is a directed graph with the set $$G$$ of vertices, and with all edges $$(u,v)$$ such that $$u\neq v$$ and $$v$$ is a power of $$u$$. For each directed graph $$D$$, we give a complete description of all groups $$G$$ such that every infinite subset of $$G$$ contains a power subgraph isomorphic to $$D$$. Also, we describe the structure of the power graphs of all finite abelian groups.
For the entire collection see [Zbl 0942.00022].

##### MSC:
 05C25 Graphs and abstract algebra (groups, rings, fields, etc.)
##### Keywords:
power graph; directed graph; finite abelian groups