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On irregular infinite Jordan groups. (English) Zbl 1277.20002

A Jordan group is a transitive permutation group \(G\) on a set \(\Omega\) which contains a subgroup \(H\) such that: (i) \(H\) acts transitively on some proper subset \(\Gamma\) of \(\Omega\) and fixes each point in \(\Omega\setminus\Gamma\); and (ii) \(|\Omega\setminus\Gamma|>k\) if \(G\) is \((k+1)\)-transitive on \(\Omega\). The current paper is part of a project of the author (with several co-authors) to classify infinite Jordan groups [see S. A. Adeleke and P. M. Neumann, Mem. Am. Math. Soc. 623 (1998; Zbl 0896.08001); J. Lond. Math. Soc., II. Ser. 53, No. 2, 230-242 (1996; Zbl 0865.20006); J. Lond. Math. Soc., II. Ser. 53, No. 2, 209-229 (1996; Zbl 0865.20005) and S. A. Adeleke and D. Macpherson, Proc. Lond. Math. Soc., III. Ser. 72, No. 1, 63-123 (1996; Zbl 0839.20002)].
The earlier work shows that most infinite Jordan groups preserve various order or betweenness relations on \(\Omega\) or preserve a nontrivial Steiner system. The author defines an infinite Jordan group \(G\) to be irregular if it is \(2\)- but not \(3\)-transitive and fails to preserve such relations (but does, necessarily, preserve the limit of such a relation). The paper defines two different types of examples of \(2\)-primitive irregular infinite Jordan groups and proves that they have the properties claimed. The author notes that he has since learnt that a similar example using a different construction is given by M. Battacharjee and D. Macpherson [in J. Group Theory 9, No. 1, 59-94 (2006; Zbl 1103.20001)].

MSC:

20B07 General theory for infinite permutation groups
20B15 Primitive groups
20B22 Multiply transitive infinite groups
20E08 Groups acting on trees
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References:

[1] Adeleke S. A., Mem. Amer. Math. Soc. 131 pp 623– (1998)
[2] Adeleke S. A., J. London Math. Soc. 53 pp 230– (1996) · Zbl 0865.20006
[3] Adeleke S. A., J. London Math. Soc. 53 pp 209– (1992) · Zbl 0865.20005
[4] Adeleke S. A., Proc. London Math. Soc. 72 pp 63– (1996) · Zbl 0839.20002
[5] Adeleke S. A., J. Combinatorial Theory 72 pp 243– (1995) · Zbl 0836.05010
[6] Battacharjee M., J. Group Theory 9 pp 59– (2006)
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[8] Droste M., Mem. Amer. Math. Soc. 57 pp 334– (1985)
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