Varieties of automorphism groups of orders

Holland, W. Charles

doi:10.1090/S0002-9947-1985-0776402-7

Varieties of automorphism groups of orders
HTML articles powered by AMS MathViewer

by W. Charles Holland PDF

Trans. Amer. Math. Soc. 288 (1985), 755-763 Request permission

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