Subgroups of Infinite Symmetric Groups

Macpherson, Dugald; Neumann, Peter M.

doi:10.1112/jlms/s2-42.1.64

Cited by 84 publications

(110 citation statements)

References 13 publications

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“…For a semigroup S, if A ⊆ S then we call the minimum cardinality of a set B such that A ∪ B = S, the relative rank of S modulo A. Alternatively, we may refer to this cardinality as the relative rank of A in S; which we denote by rank(S : A). This subject has been studied in the context of groups in [2] and [14]. In these papers, so called large subgroups of the symmetric group S X , over an infinite set X, were considered.…”

Section: Introductionmentioning

confidence: 99%

Generating the Full Transformation Semigroup Using Order Preserving Mappings

Higgins¹,

Mitchell

Ruškuc³

2003

Glasgow Math. J.

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Abstract. For a linearly ordered set X we consider the relative rank of the semigroup of all order preserving mappings O X on X modulo the full transformation semigroup T X . In other words, we ask what is the smallest cardinality of a set A of mappings such that O X ∪ A = T X . When X is countably infinite or well-ordered (of arbitrary cardinality) we show that this number is one, while when X = ‫ޒ‬ (the set of real numbers) it is uncountable.2000 Mathematics Subject Classification. 20M20, 06A05.

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Section: Introductionmentioning

confidence: 99%

Generating the Full Transformation Semigroup Using Order Preserving Mappings

Higgins¹,

Mitchell

Ruškuc³

2003

Glasgow Math. J.

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“…A deep theorem of Macpherson and Neumann [16] states that if the symmetric group Sym(κ) consisting of all permutations of a cardinal κ can be written as a union of an increasing chain G i : i < λ of proper subgroups G i , then λ > κ. Throughout this paper the minimal λ with this property will be denoted by cf(Sym(κ)).…”

Section: Introductionmentioning

confidence: 99%

Measurable cardinals and the cofinality of the symmetric group

Friedman

Zdomskyy

2010

Fund. Math.

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Abstract. Assuming the existence of a P 2 κ-hypermeasurable cardinal, we construct a model of Set Theory with a measurable cardinal κ such that 2 κ = κ ++ and the group Sym(κ) of all permutations of κ cannot be written as a union of a chain of proper subgroups of length < κ ++ . The proof involves the iteration of a suitably defined uncountable version of the Miller forcing poset as well as the "tuning fork" argument introduced by the first author and K. Thompson in [11].

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“…So there is a finite set B such that Aut(R), B = Sym(R). (This fact follows from Theorem 1.1 of [43]. Galvin [30] showed that we can take B to consist of a single element.…”

Section: Theorem 210mentioning

confidence: 70%

Aspects of infinite permutation groups

Cameron

2007

Groups St Andrews 2005

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Until 1980, there was no such subgroup as 'infinite permutation groups', according to the Mathematics Subject Classification: permutation groups were assumed to be finite. There were a few papers, for example [10,62], and a set of lecture notes by Wielandt [72], from the 1950s. Now, however, there are far more papers on the topic than can possibly be summarised in an article like this one.I shall concentrate on a few topics, following the pattern of my conference lectures: the random graph (a case study); homogeneous relational structures (a powerful construction technique for interesting permutation groups); oligomorphic permutation groups (where the relations with other areas such as logic and combinatorics are clearest, and where a number of interesting enumerative questions arise); and the Urysohn space (another case study). I have preceded this with a short section introducing the language of permutation group theory, and I conclude with briefer accounts of a couple of topics that didn't make the cut for the lectures (maximal subgroups of the symmetric group, and Jordan groups).I have highlighted a few specific open problems in the text. It will be clear that there are many wide areas needing investigation! Notation and terminologyThis section contains a few standard definitions concerning permutation groups. I write permutations on the right: that is, if g is a permutation of a set Ω, then the image of α under g is written αg.The symmetric group Sym(Ω) on a set Ω is the group consisting of all permutations of Ω. If Ω is infinite and c is an infinite cardinal number not exceeding Ω, the bounded symmetric group BSym c (Ω) consists of all permutations moving fewer than c points; if c = ℵ 0 , this is the finitary symmetric group FSym(Ω) consisting of all finitary permutations (moving only finitely many points). The alternating group Alt(Ω) is the group of all even permutations, where a permutation is even if it moves only finitely many points and acts as an even permutation on its support.Assuming the Axiom of Choice, the only nontrivial normal subgroups of Sym(Ω) for an infinite set Ω are the bounded symmetric groups and the alternating group.A permutation group on a set Ω is a subgroup of the symmetric group on Ω. As noted above, we denote the image of α under the permutation g by αg. For the most part, I will be concerned with the case where Ω is countably infinite.

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Subgroups of Infinite Symmetric Groups

Cited by 84 publications

References 13 publications

Generating the Full Transformation Semigroup Using Order Preserving Mappings

Generating the Full Transformation Semigroup Using Order Preserving Mappings

Measurable cardinals and the cofinality of the symmetric group

Aspects of infinite permutation groups

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