A combinatorial property and power graphs of groups. (English) Zbl 0966.05040

Kelarev, A. V.; Quinn, S. J.

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].

Reviewer: Steven C.Althoen (Flint)

MSC:

05C25

Graphs and abstract algebra (groups, rings, fields, etc.)

Keywords:

power graph; directed graph; finite abelian groups