Freiman isomorphism

Let A and B be subsets of abelian groupsMathworldPlanetmath GA and GB respectively. A Freiman isomorphism of order s is a bijectiveMathworldPlanetmathPlanetmath mapping f:AB such that


holds if and only if


The Freiman isomorphism is a restrictionPlanetmathPlanetmathPlanetmath of the conventional notion of a group isomorphism to a limited number of group operationsMathworldPlanetmath. 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 isomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath.

Freiman isomorphisms were introduced by Freiman in his monograph [1] to build a general theory of set additionPlanetmathPlanetmath ( that is independent of the underlying group.

The number of equivalence classesMathworldPlanetmathPlanetmath of n-element sets of integers under Freiman isomorphisms of order 2 is n2n(1+o(1)) [2].


  • 1 Gregory Freiman. Foundations of Structural Theory of Set Addition, volume 37 of TranslationsMathworldPlanetmathPlanetmath of Mathematical Monographs. AMS, 1973. 0271.10044.
  • 2 Sergei V. Konyagin and Vsevolod F. Lev. Combinatorics and linear algebra of Freiman’s isomorphism. Mathematika, 47:39–51, 2000. Available at seva/pub_list.html seva/.
  • 3 Melvyn B. Nathanson. Additive Number Theory: Inverse Problems and Geometry of Sumsets, volume 165 of GTM. Springer, 1996. 0859.11003.
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