Automorphism groups of countable highly homogeneous partially ordered sets. (English) Zbl 0792.20002
We prove Theorem 1. The automorphism group \(G\) of the countable universal poset is simple. Actually, if \(g\in G\) and \(g \neq 1\), then every element of \(G\) can be written in the form \(\prod^ 8_{j = 1}(h^{-1}_{2j-1} gh_{2j-1} \cdot h^{-1}_{2j} g^{-1} h_{2j})\), for some \(h_ 1,\dots,h_{16} \in G\). Theorem 2. If \((\Omega,\leq)\) is a countable highly homogeneous poset, then “almost all” finitely generated subgroups of \(\text{Aut}(\Omega,\leq)\) are free. (“Almost all” is in the Baire Category sense).
Reviewer: A.M.W.Glass (Bowling Green)
MSC:
| 20B27 | Infinite automorphism groups |
| 20F05 | Generators, relations, and presentations of groups |
| 06A06 | Partial orders, general |
| 20E07 | Subgroup theorems; subgroup growth |
| 20E32 | Simple groups |
| 20B22 | Multiply transitive infinite groups |
Keywords:
automorphism group; countable universal poset; highly homogeneous poset; finitely generated subgroupsReferences:
| [1] | Chang, C.C., Keisler, H.J.: Model Theory. (Stud. Logic, vol. 73) Amsterdam: N. Holland 1973 · Zbl 0276.02032 |
| [2] | Dixon, J.D.: Most finitely generated permutation groups are free. Bull. Lond. Math. Soc.22, 222–226 (1990) · Zbl 0675.20003 · doi:10.1112/blms/22.3.222 |
| [3] | Dixon, J.D., Neumann, P.M., Thomas, S.: Subgroups of small index in finite symmetric groups. Bull Lond. Math. Soc.18, 580–586 (1986) · Zbl 0607.20003 · doi:10.1112/blms/18.6.580 |
| [4] | Droste, M.: Structure of partially ordered sets with transitive automorphism groups. Mem. Am. Math. Soc.334 (1985) · Zbl 0574.06001 |
| [5] | Droste, M., Truss, J.K.: Subgroups of small index in ordered permutation groups. Q.J. Math. Oxf.42, 31–47 (1991) · Zbl 0727.06014 · doi:10.1093/qmath/42.1.31 |
| [6] | Glass, A.M.W.: Ordered Permutation Groups. (Lond. Math Soc. Lect. Notes Ser., vol 55) Cambridge: Cambridge University Press 1981 · Zbl 0473.06010 |
| [7] | Glass, A.M.W.: The Ubiquity of Free Groups. Math. Intell.14, 54–57 (1992) · Zbl 0791.20001 · doi:10.1007/BF03025870 |
| [8] | Glass, A.M.W., McCleary, S.H.: Highly transitive representations of free groups and free products. Bull. Australian Math. Soc.43, 19–36 (1991) · Zbl 0717.20001 · doi:10.1017/S0004972700028744 |
| [9] | Hodges, W., Hodkinson, I., Lascar, D., Shelah, S.: The small index property for {\(\omega\)}-stable {\(\omega\)}-categorical structures and for the random graph. (Preprint) · Zbl 0788.03039 |
| [10] | Hrushovski, E.: Extending partial automorphisms of graphs (to appear) · Zbl 0767.05053 |
| [11] | Karrass, A., Solitar, D.: Some remarks on the infinite symmetric group. Math. Z.66, 64–69 (1956) · Zbl 0071.02401 · doi:10.1007/BF01186596 |
| [12] | Macpherson, H.D.: Groups of automorphisms of 0-categorical structures. Q. J. Math. Oxf.37, 449–465 (1986) · Zbl 0611.03014 · doi:10.1093/qmath/37.4.449 |
| [13] | Neumann, B.H.: On Ordered Groups, Am. J. Math.71, 1–18 (1949) · Zbl 0031.34201 · doi:10.2307/2372087 |
| [14] | Rubin, M.: Simple countable structures. (Preprint) |
| [15] | Schmerl, J.H.: Countable homogeneous partially ordered sets. Algebra Univers.9, 317–321 (1979) · Zbl 0423.06002 · doi:10.1007/BF02488043 |
| [16] | Truss, J.K.: The group of the countable universal graph. Math. Proc. Camb. Philos. Soc.98, 213–245 (1985) · Zbl 0586.20004 · doi:10.1017/S0305004100063428 |
| [17] | Truss, J.K.: Infinite permutation groups: subgroups of small index. J. Algebra120, 495–515 (1989) · Zbl 0666.20002 |
| [18] | Truss, J.K.: Infinite simple permutation groups–a survey. In: Campbell, C.M., Robertson, E.F. (eds.) Groups. St. Andrews 1989, vol. 2. (Lond. Math. Soc. Lect. Notes Ser. vol. 160) Cambridge: Cambridge University Press 1991 · Zbl 0790.20003 |
| [19] | White, S.: The group generated byx+1 andx p is free. J. Algebra118, 408–422 (1988) · Zbl 0662.20024 · doi:10.1016/0021-8693(88)90030-0 |
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.
