Adeleke, S. A. On irregular infinite Jordan groups. (English) Zbl 1277.20002 Commun. Algebra 41, No. 4, 1514-1546 (2013). 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)]. Reviewer: John D. Dixon (Ottawa) Cited in 3 Documents MSC: 20B07 General theory for infinite permutation groups 20B15 Primitive groups 20B22 Multiply transitive infinite groups 20E08 Groups acting on trees Keywords:transitive permutation groups; infinite Jordan groups; betweenness relations; semilinear orders Citations:Zbl 0896.08001; Zbl 0865.20006; Zbl 0865.20005; Zbl 0839.20002; Zbl 1103.20001 PDFBibTeX XMLCite \textit{S. A. Adeleke}, Commun. Algebra 41, No. 4, 1514--1546 (2013; Zbl 1277.20002) Full Text: DOI 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) [7] Cameron P. J., Math. Z. 148 pp 127– (1976) · Zbl 0313.20022 [8] Droste M., Mem. Amer. Math. Soc. 57 pp 334– (1985) This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.