Infinite 32‐generated groups

Infinite Groups

Most interesting groups arise as a group of transformations. For example the set of “rigid motions” of 2-dimensional space R forms a group denoted Isom(R). There are three kinds of transformations: reflection about a line, rotation about a point, and translation in a direction. These form a group under composition (i.e., do one transformation and then do the other). For example, the composition...

Infinite Permutation Groups

Groups with Infinite Homology

A group is said to be acyclic if its reduced homology vanishes. Many interesting classes of groups have been discovered having this property ([2] is a useful survey). This is an indication that the reduced homology carries limited information. Here we obtain information about H̃(G) for a class of groups G that includes all locally finite groups and all soluble groups of finite rank. We show that...

Infinite Bounded Permutation Groups

This work concerns permutation groups on uncountable sets. The environment is ZFC, Set Theory with Axiom of Choice. Therefore the cardinal numbers are wellordered, a cardinal number k is infinite if and only if Ko < k, and each cardinal number k has a unique successor k. Throughout the paper Cl will be a set (usually infinite), n its cardinality, and m a cardinal number less than n. The support...

Generating infinite symmetric groups

Let S = Sym(Ω) be the group of all permutations of an infinite set Ω. Extending an argument of Macpherson and Neumann, it is shown that if U is a generating set for S as a group, respectively as a monoid, then there exists a positive integer n such that every element of S may be written as a group word, respectively a monoid word, of length ≤ n in the elements of U. Some related questions are n...