ORDERED GROUPS IN WHICH ALL CONVEX JUMPS ARE CENTRAL

Title & Authors
ORDERED GROUPS IN WHICH ALL CONVEX JUMPS ARE CENTRAL
Bludov, V.V.; Glass, A.M.W.; Rhemtulla, Akbar H.;

Abstract
(G, <) is an ordered group if'<'is a total order relation on G in which f < g implies that xfy < xgy for all f, g, x, y $\small{\in}$ G. We say that (G, <) is centrally ordered if (G, <) is ordered and ［G,D］ $\small{\subseteq}$ C for every convex jump C $\small{\prec}$ D in G. Equivalently, if $\small{f^{-1}g f{\leq} g^2}$ for all f, g $\small{\in}$ G with g > 1. Every order on a torsion-free locally nilpotent group is central. We prove that if every order on every two-generator subgroup of a locally soluble orderable group G is central, then G is locally nilpotent. We also provide an example of a non-nilpotent two-generator metabelian orderable group in which all orders are central.
Keywords
soluble group;locally nilpotent group;ordered group;convex jump;central series;weakly Abelian;
Language
English
Cited by
1.
On centrally orderable groups, Journal of Algebra, 2005, 291, 1, 129
2.
On SolvableR* Groups of Finite Rank, Communications in Algebra, 2003, 31, 7, 3287
References
1.
Series in Algebra, 1999. vol.7.

2.
Proc. London Math. Soc., 1957. vol.7. pp.29-62

3.
Proc. London Math. Soc., 1954. vol.4. pp.420-436

4.
Lectures given at the Canadian Mathematical Congress, 1957.

5.
Proc. London Math. Soc., 1966. vol.16. pp.1-39

6.
J. Aust. Math. Soc. (Series A), 1988. vol.45. pp.296-302

7.
Algebra & Logic, 1968. vol.7. pp.160-161

8.
Fully Ordered Groups, 1974.

9.
Algebra & Logic, 1998. vol.37. pp.170-180

10.
The Theory of Lattice-ordered Groups, 1994.

11.
Proc. Amer. Math. Soc., 1952. vol.3. pp.579-583

12.
Math. Z., 1974. vol.137. pp.265-284

13.
Lecture Notes in Pure & Appl. Math., 1977. vol.27.

14.
Osaka J. Math., 1950. vol.2. pp.161-164

15.
Czechoslovak Math. J., 1983. vol.33. pp.348-353

16.
Proc. Amer. Math. Soc., 1973. vol.41. pp.31-33

17.
Grad. Texts in Math., 1995. vol.80.

18.
Ucen. Zap. Ivanovs. Gos. Ped. Inst., 1953. vol.4. pp.92-96

19.
J. Algebra, 1972. vol.20. pp.250-270