Abstract
A right-ordered group is any subgroup of a group of order preserving permutations of some totally ordered set. It can also be characterized as a group G together with a total order relation ≤ such that a ≤ b implies ac ≤ bc for all a,b,c in G. It is not known whether all such groups G have a series from {e} to G with torsion-free abelian factors. Here we show that a polycyclic right-ordered group must have such a series. This characterizes polycyclic right-ordered groups as the class of poly infinite-cyclic groups.
Keywords
- Normal Subgroup
- Polycyclic Group
- Convex Subgroup
- Maximal Normal Subgroup
- Engel Group
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References
P. F. Conrad, "Right-ordered groups", Michigan Math. J. 6(1959), 267–275.
R. B. Mura and A. H. Rhemtulla, Orderable Groups, Lecture notes in pure and applied mathematics. Vol. 27(1977), Marcel Dekker Inc., New York.
D. M. Smirnov, "Right-ordered groups", Algebra i Logika 5:6(1966), 41–59.
D. J. S. Robinson, Finiteness conditions and generalized soluble groups, 1,2(1972), Springer-Verlag, New York.
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© 1981 Springer-Verlag
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Rhemtulla, A.H. (1981). Polycyclic right-ordered groups. In: Amayo, R.K. (eds) Algebra Carbondale 1980. Lecture Notes in Mathematics, vol 848. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0090569
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DOI: https://doi.org/10.1007/BFb0090569
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