On permutation groups with bounded movement. (English) Zbl 0744.20004
Author’s abstract: “Let \(G\) be a permutation group on a set \(\Omega\) with no fixed points in \(\Omega\) and let \(m\) be a positive integer. If no element of \(G\) moves any subset of \(\Omega\) by more than \(m\) points (that is if \(| \Gamma-\Gamma^ g|\leq m\) for every \(\Gamma\subseteq\Omega\) and \(g\in G\)), then \(\Omega\) is finite, and moreover \(|\Omega |\leq 5m-2\). If moreover \(G\) is transitive on \(\Omega\) then \(| \Omega|\leq 3m\) and equality can be achieved if \(m=2\) or \(m\) is a power of 3”.
Reviewer: J.Libicher (Brno)
MSC:
| 20B10 | Characterization theorems for permutation groups |
| 20B05 | General theory for finite permutation groups |
| 20B20 | Multiply transitive finite groups |
References:
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| [2] | Fein, B.; Kantor, W. M.; Schacher, M., Relative Brauer groups, II, J. reine angew. Math., 328, 39-57 (1981) · Zbl 0457.13004 |
| [3] | Neumann, B. H., Groups covered by permutable subsets, J. London Math. Soc., 29, 236-248 (1954) · Zbl 0055.01604 |
| [4] | Neumann, P. M., The structure of finitary permutation groups, Arch. Math. (Basel), 29, 3-17 (1976) · Zbl 0324.20037 |
| [5] | Rotman, J. J., The Theory of Groups: An Introduction (1965), Allyn & Bacon: Allyn & Bacon Boston · Zbl 0123.02001 |
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