Varieties of automorphism groups of orders
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- by W. Charles Holland
- Trans. Amer. Math. Soc. 288 (1985), 755-763
- DOI: https://doi.org/10.1090/S0002-9947-1985-0776402-7
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Abstract:
The group $A(\Omega )$ of automorphisms of a totally ordered set $\Omega$ must generate either the variety of all groups or the solvable variety of class $n$. In the former case, $A(\Omega )$ contains a free group of rank ${2^{{\aleph _0}}}$; in the latter case, $A(\Omega )$ contains a free solvable group of class $n - 1$ and rank ${2^{{\aleph _0}}}$.References
- C. C. Chang and A. Ehrenfeucht, A characterization of abelian groups of automorphisms of a simply ordering relation, Fund. Math. 51 (1962/63), 141–147. MR 142629, DOI 10.4064/fm-51-2-141-147
- A. M. W. Glass, Ordered permutation groups, London Mathematical Society Lecture Note Series, vol. 55, Cambridge University Press, Cambridge-New York, 1981. MR 645351
- A. M. W. Glass, Yuri Gurevich, W. Charles Holland, and Saharon Shelah, Rigid homogeneous chains, Math. Proc. Cambridge Philos. Soc. 89 (1981), no. 1, 7–17. MR 591966, DOI 10.1017/S0305004100057881
- A. M. W. Glass, W. Charles Holland, and Stephen H. McCleary, The structure of $l$-group varieties, Algebra Universalis 10 (1980), no. 1, 1–20. MR 552151, DOI 10.1007/BF02482885
- W. Charles Holland, The largest proper variety of lattice ordered groups, Proc. Amer. Math. Soc. 57 (1976), no. 1, 25–28. MR 406902, DOI 10.1090/S0002-9939-1976-0406902-0
- Jorge Martinez, Varieties of lattice-ordered groups, Math. Z. 137 (1974), 265–284. MR 354483, DOI 10.1007/BF01214370
- Stephen H. McCleary, The structure of intransitive ordered permutation groups, Algebra Universalis 6 (1976), no. 2, 229–255. MR 424638, DOI 10.1007/BF02485831
- Jan Mycielski, Almost every function is independent, Fund. Math. 81 (1973), no. 1, 43–48. MR 339091, DOI 10.4064/fm-81-1-43-48
- Hanna Neumann, Varieties of groups, Springer-Verlag New York, Inc., New York, 1967. MR 0215899
- Tadashi Ohkuma, Sur quelques ensembles ordonnés linéairement, Fund. Math. 43 (1956), 326–337 (French). MR 84486
- John A. Read, Wreath products of nonoverlapping lattice ordered groups, Canad. Math. Bull. 17 (1974/75), no. 5, 713–722. MR 384642, DOI 10.4153/CMB-1974-129-8
Bibliographic Information
- © Copyright 1985 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 288 (1985), 755-763
- MSC: Primary 06F15; Secondary 08B20, 20B27, 20E10, 20F16
- DOI: https://doi.org/10.1090/S0002-9947-1985-0776402-7
- MathSciNet review: 776402