Factorizations of groups and related topics

Abstract

This is a survey of some recent progress in the theory of groups with factorizations. Some of the methods can be used to obtain information about finite groups in general, nilpotent algebras and nearrings.

Factorizations of Self-similar Groups Associated to the First Grigorchuk Group

“…Factorizations are studied intensively (for a recent survey see, for example, [2,4]). The connection between properties of a factorizable group and its factors are not very clear and obvious in general which is a source of many problems.…”

“…Many authors use the name "binary adding machine" to any conjugate to A rather than to A itself 2. After Frank Gray; it is also called a "reflected binary code" (this name was introduced by the author in his patent application in 1947).Downloaded by [Athabasca University] at 11:21 24 June 2016…”

A note on infinite aS-groups

“…An important class of factorizable groups are groups which have a triple factorization. For more information on factorizable groups and triple factorization we refer the reader to [1], [2], [4], [7] and [9].…”

On Factorizable Semihypergroups

“…The decomposition property of a set appears as a key element in many topics in algebra, being connected, for example, with equivalence relations (and called partition), factorizable semigroups [1,2], factorization of groups [3], breakable semigroups [4], or recently introduced breakable semihypergroups [5]. A group G is called factorized if it can be written as a product of two subgroups A and B.…”

“…A group G is called factorized if it can be written as a product of two subgroups A and B. This means that any element g in G has the form g = ab for some a ∈ A and b ∈ B [3]. One of the most famous results about factorization of groups, proven in 1955 by Ito [6], concerns the product of two abelian subgroups.…”