Freiman isomorphism

Let $A$ and $B$ be subsets of abelian groups $G_A$ and $G_B$ respectively. A Freiman isomorphism of order $s$ is a bijective mapping $f:A\to B$ such that

$$a_1+a_2+\mathrm{\cdots }+a_s=a_1^{\prime }+a_2^{\prime }+\mathrm{\cdots }+a_s^{\prime }$$

holds if and only if

$$f(a_1)+f(a_2)+\mathrm{\cdots }+f(a_s)=f(a_1^{\prime })+f(a_2^{\prime })+\mathrm{\cdots }+f(a_s^{\prime }).$$

The Freiman isomorphism is a restriction of the conventional notion of a group isomorphism to a limited number of group operations. In particular, a Freiman isomorphism of order $s$ is also a Freiman isomorphism of order $s-1$, and the mapping is a Freiman isomorphism of every order precisely when it is the conventional isomorphism.

Freiman isomorphisms were introduced by Freiman in his monograph [1] to build a general theory of set addition (http://planetmath.org/Sumset) that is independent of the underlying group.

The number of equivalence classes of $n$-element sets of integers under Freiman isomorphisms of order $2$ is $n^{2n(1+o(1))}$ [2].

Title	Freiman isomorphism
Canonical name	FreimanIsomorphism
Date of creation	2013-03-22 13:40:39
Last modified on	2013-03-22 13:40:39
Owner	bbukh (348)
Last modified by	bbukh (348)
Numerical id	8
Author	bbukh (348)
Entry type	Definition
Classification	msc 11B75
Classification	msc 20K30
Related topic	Isomorphism2