Maximal Subgroups of Infinite Symmetric Groups

1967 ◽  
Vol 10 (3) ◽  
pp. 375-381 ◽  
Author(s):  
Fred Richman
Keyword(s):  
Boolean Algebra ◽  
Symmetric Groups ◽  
Discrete Space ◽  
Maximal Subgroups ◽  
Infinite Set

The purpose of this paper is to extend results of Ball [1] concerning maximal subgroups of the group S(X) of all permutations of the infinite set X. The basic idea is to consider S(X) as a group of operators on objects more complicated than X. The objects we consider here are subspaces of the Stone-Čech compactification of the discrete space X and the Boolean algebra of “big setoids” of X.

SOME MAXIMAL SUBGROUPS OF INFINITE SYMMETRIC GROUPS

1996 ◽  
Vol 47 (3) ◽  
pp. 297-311 ◽  
Author(s):  
JACINTA COVINGTON ◽  
DUGALD MACPHERSON ◽  
ALAN MEKLER
Keyword(s):  
Symmetric Groups ◽  
Maximal Subgroups

Maximal Subgroups of Infinite Symmetric Groups

1990 ◽  
Vol s2-42 (1) ◽  
pp. 85-92 ◽  
Author(s):  
H. D. Macpherson ◽  
Cheryl E. Praeger
Keyword(s):  
Symmetric Groups ◽  
Maximal Subgroups

t-Representability of maximal subgroups of symmetric groups

2018 ◽  
Vol 9 (1) ◽  
pp. 27
Author(s):  
Sini P
Keyword(s):  
Symmetric Group ◽  
Symmetric Groups ◽  
Maximal Subgroups ◽  
Group Of Homeomorphisms

A subgroup \(H\) of the group \(S(X)\) of all permutations of a set \(X\) is called \(t\)−representable on \(X\) if there exists a topology \(T\) on \(X\) such that the group of homeomorphisms of \((X, T ) = K\). In this paper we study the \(t\)-representability of maximal subgroups of the symmetric group.

Recursive isomorphism types of recursive Boolean algebras

1981 ◽  
Vol 46 (3) ◽  
pp. 572-594 ◽  
Author(s):  
J. B. Remmel
Keyword(s):  
Boolean Algebra ◽  
Recursive Function ◽  
Boolean Algebras ◽  
Isomorphism Type ◽  
Turing Degrees ◽  
Main Question ◽  
Partial Recursive Function ◽  
Recursive Isomorphism ◽  
Infinite Set ◽  
The Ideal

A Boolean algebra is recursive if B is a recursive subset of the natural numbers N and the operations ∧ (meet), ∨ (join), and ¬ (complement) are partial recursive. Given two Boolean algebras and , we write if is isomorphic to and if is recursively isomorphic to , that is, if there is a partial recursive function f: B1 → B2 which is an isomorphism from to . will denote the set of atoms of and () will denote the ideal generated by the atoms of .One of the main questions which motivated this paper is “To what extent does the classical isomorphism type of a recursive Boolean algebra restrict the possible recursion theoretic properties of ?” For example, it is easy to see that must be co-r.e. (i.e., N − is an r.e. set), but can be immune, not immune, cohesive, etc? It follows from a result of Goncharov [4] that there exist classical isomorphism types which contain recursive Boolean algebras but do not contain any recursive Boolean algebras such that is recursive. Thus the classical isomorphism can restrict the possible Turing degrees of , but what is the extent of this restriction? Another main question is “What is the recursion theoretic relationship between and () in a recursive Boolean algebra?” In our attempt to answer these questions, we were led to a wide variety of recursive isomorphism types which are contained in the classical isomorphism type of any recursive Boolean algebra with an infinite set of atoms.

Normal subgroups of infinite symmetric groups, with an application to stratified set theory

2009 ◽  
Vol 74 (1) ◽  
pp. 17-26 ◽  
Author(s):  
Nathan Bowler ◽  
Thomas Forster
Keyword(s):  
Set Theory ◽  
Symmetric Groups ◽  
Axiom Of Choice ◽  
Normal Subgroups ◽  
Standard Analysis ◽  
Small Index ◽  
The Axiom Of Choice ◽  
Two Stages ◽  
Infinite Set ◽  
Stratified Set

It is generally known that infinite symmetric groups have few nontrivial normal subgroups (typically only the subgroups of bounded support) and none of small index. (We will explain later exactly what we mean by small). However the standard analysis relies heavily on the axiom of choice. By dint of a lot of combinatorics we have been able to dispense—largely—with the axiom of choice. Largely, but not entirely: our result is that if X is an infinite set with ∣X∣ = ∣X × X∣ then Symm(X) has no nontrivial normal subgroups of small index. Some condition like this is needed because of the work of Sam Tarzi who showed [4] that, for any finite group G, there is a model of ZF without AC in which there is a set X with Symm(X)/FSymm(X) isomorphic to G.The proof proceeds in two stages. We consider a particularly useful class of permutations, which we call the class of flexible permutations. A permutation of X is flexible if it fixes at least ∣X∣-many points. First we show that every normal subgroup of Symm(X) (of small index) must contain every flexible permutation. This will be theorem 4. Then we show (theorem 7) that the flexible permutations generate Symm(X).

Maximal Subgroups of Infinite Symmetric Groups

1994 ◽  
Vol s3-68 (1) ◽  
pp. 77-111 ◽  
Author(s):  
Marcus Brazil ◽  
Jacinta Covington ◽  
Tim Penttila ◽  
Cheryl E. Praeger ◽  
Alan R. Woods
Keyword(s):  
Symmetric Groups ◽  
Maximal Subgroups
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