Ordered Permutation Groups

Glass, A. M. W.

doi:10.1017/cbo9780511721243

Cited by 29 publications

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“…§2.3 of [10]). We will now examine the structure of (Jf, ç ) for arbitrary cofinality and coinitiality of ß.…”

Section: Characterization Of J/~(a(ü)mentioning

confidence: 98%

“…Here we make use of the fact that A(ü) is an /-group. For the techniques used in the proofs of 3.4-3.10 we refer the reader to §2.2 of Glass [10]. Given aje A(ü), aß denotes ß~laß.…”

Section: Characterization Of J/~(a(ü)mentioning

confidence: 99%

“…the group of all order-preserving permutations) A($i) acts 2-transitively on it. Chains (ß, <) of this type and their automorphism groups A(Q) have been used for the construction of infinite simple torsion-free groups (Higman [12]) or, in the theory of lattice-ordered groups (/-groups), in dealing with embeddings of arbitrary /-groups into simple divisible /-groups (Holland [13]) (for a variety of further results see Glass [10]). …”

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confidence: 99%

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Normal Subgroups of Doubly Transitive Automorphism Groups of Chains

Ball¹,

Droste²

1985

Transactions of the American Mathematical Society

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Abstract.We characterize the structure of the normal subgroup lattice of 2-transitive automorphism groups A(Q) of infinite chains (ß, <) by the structure of the Dedekind completion (Í2, <) of the chain (Q, <). As a consequence we obtain various group-theoretical results on the normal subgroups of A(Q), including that any proper subnormal subgroup of A(Q) is indeed normal and contained in a maximal proper normal subgroup of A(Q), and that A(Q) has precisely 5 normal subgroups if and only if the coterminality of the chain (Ü, <) is countable.

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“…§2.3 of [10]). We will now examine the structure of (Jf, ç ) for arbitrary cofinality and coinitiality of ß.…”

Section: Characterization Of J/~(a(ü)mentioning

confidence: 98%

“…Here we make use of the fact that A(ü) is an /-group. For the techniques used in the proofs of 3.4-3.10 we refer the reader to §2.2 of Glass [10]. Given aje A(ü), aß denotes ß~laß.…”

Section: Characterization Of J/~(a(ü)mentioning

confidence: 99%

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confidence: 99%

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Normal Subgroups of Doubly Transitive Automorphism Groups of Chains

Ball¹,

Droste²

1985

Transactions of the American Mathematical Society

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“…As is standard, this makes W into an ℓ-group (see [3], Chapter 5) with the cardinal direct sum order on B.…”

Section: Background and Notationmentioning

confidence: 99%

Finitely generated lattice-ordered groups with soluble word problem

Glass¹

2008

Journal of Group Theory

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Finitely generated lattice-ordered groups with soluble word problem. A. M. W. Glass Submitted 19th March 2007 AbstractWilliam W. Boone and Graham Higman proved that a finitely generated group has soluble word problem if and only if it can be embedded in a simple group that can be embedded in a finitely presented group. We prove the exact analogue for lattice-ordered groups:Theorem: A finitely generated lattice-ordered group has soluble word problem if and only if it can be ℓ-embedded in an ℓ-simple lattice-ordered group that can be ℓ-embedded in a finitely presented lattice-ordered group.The proof uses permutation groups, a technique of Holland and McCleary, and the ideas used to prove the lattice-ordered group analogue of Higman's Embedding Theorem.

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“….)). As in [4], we shall write A(X) throughout for the automorphism group of the chain (linearly ordered set) (X, <), and also if X is a coloured chain. In this paper we write group actions on the left, and for any permutation group G acting on , we write G X and G {X} for the pointwise and setwise stabilizers of X ⊆ in G respectively.…”

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confidence: 99%

The Small Index Property for Countable 1-Transitive Linear Orders

Chicot¹,

Truss²

2005

Glasgow Math. J.

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Abstract. It is shown that the countable saturated discrete linear ordering has the small index property, but that the countable 1-transitive linear orders which contain a convex subset isomorphic to ‫ޚ‬ 2 do not. Similar results are also proved in the coloured case.2000 Mathematics Subject Classification. 20B27, 06F15.1. Introduction. In [9] it was shown that the ordered set of rational numbers has the 'small index property' SIP, meaning that any subgroup of its automorphism group having index strictly less than 2 ℵ 0 contains the pointwise stabilizer of a finite set. The small index property has received a great deal of attention in quite a wide variety of special cases. Its model-theoretic significance is that its truth tells us that the natural topological group associated with the structure (under the topology of pointwise convergence) can be recovered from the pure group, and from this one can deduce that the structure is interpretable in the (abstract) automorphism group [5].A conjecture of Macpherson [7] was that the SIP should hold for every ℵ 0 -categorical structure. This was refuted by Hrushovski, but the conjecture remains in modified form (see [6], bottom of page 52).In this paper we look at a class of countable structures which are (mostly) not ℵ 0 -categorical, namely the countable '1-transitive' linear orders classified by Morel [8]. (A linear order X is 1-transitive if its automorphism group acts (singly) transitively on X.) These have arisen as building blocks for various other classes of countable structures, in particular for certain 'cycle-free' partial orders [11], and in [10] the small index property was investigated for some of these structures. Generally the SIP was established there for structures built using only the very simplest of Morel's cases, and it was left open as to whether it might hold more generally. What was wanted was to find for which of Morel's structures the SIP held.Rather disappointingly, we are only able to establish the SIP for ‫ޑ‬ (already known), ‫ޚ‬ (trivial), and ‫ޚ.ޑ‬ (new). For all Morel's other orders, ‫ޚ‬ α and ‫ޚ.ޑ‬ α for ordinals α ≥ 2, the SIP is definitely false. Meanwhile, Duby [3] examined the coloured case, and was able to establish the SIP for all the ℵ 0 -categorical coloured orders among those given in [1, 2] (essentially those which only have finitely many colours, and which contain no discrete orderings in their coding trees). In view of the example of ‫ޚ.ޑ‬ (which we already had shown has the SIP) he asked whether one could establish the SIP for all the saturated structures among those classified, and we answer this affirmatively here.

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Ordered Permutation Groups

Cited by 29 publications

References 0 publications

Normal Subgroups of Doubly Transitive Automorphism Groups of Chains

Normal Subgroups of Doubly Transitive Automorphism Groups of Chains

Finitely generated lattice-ordered groups with soluble word problem

The Small Index Property for Countable 1-Transitive Linear Orders

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