Non-abelian ordered groups
Paul F. Conrad · Pacific Journal of Mathematics · 1959
l Introduction* In this paper we prove some theorems about nonabelian o-groups, and give some methods of constructing such groups.Most of the literature on o-groups is concerned with abelian o-groups, and the examples in print of non-abelian o-groups are few.Iwasawa [8] proves that any free group can be ordered, and he also gives some additional examples of o-groups.Vinogradov [15] shows that the free product of two o-groups A and B can be ordered so as to preserve the given orders.Chehata [1] gives an example of an o-group that is simple.[3] and [11] contain examples of o-groups.Most of the theorems in this paper give methods for constructing o-groups.For example, in §3 we study the o-automorphisms of an o-group G.For every group A of o-automorphisms of G that can be ordered we can construct a new o-group H that contains A and G. H is the natural splitting extension of G by A. In § 5 the relationship between central extensions and bilinear mappings is exploited.It is shown that any skew-symmetric real matrix can be used to construct o-groups.In §6 some o-groups of rank 2 are constructed.In § 4 a study is made of the ordered extensions of a subgroup of the reals.One of the main results is a necessary and sufficient condition for such an extension to split.The principal tool used throughout is the extension theory of Schreier [14].2. Notation and Terminology.The notation of [3] is used throughout.In particular, the notation and results from § 2 [3, pp.517-518] are used repeatedly.Unless otherwise stated the group operation will always be addition and 0 will denote a group identity.N and N f are o-groups with elements α, 6, c, and α', b f , c', respectively.G is a normal o-extension of N by JV\ We identify G with its representation G' = N' x N, whereand (α', α) is positive if a! > 0 or a! = 0 and a > 0. See [3] for the properties of the factor mapping / and the representative function r. θ will always denote a trivial homomorphism of a group onto the identity element of some other group.For an o-group H, let A(H) be the group of all o-automorphisms of H.For an abelian o-group K, let D{K) be the ^-closure or completion of K.In particular, D(K) is a vector space over the rationale and there is a natural extension of the order