# 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 $f\colon A\to B$ such that

 $a_{1}+a_{2}+\cdots+a_{s}=a^{\prime}_{1}+a^{\prime}_{2}+\cdots+a^{\prime}_{s}$

holds if and only if

 $f(a_{1})+f(a_{2})+\cdots+f(a_{s})=f(a^{\prime}_{1})+f(a^{\prime}_{2})+\cdots+f% (a^{\prime}_{s}).$

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].

## References

• 1 Gregory Freiman. Foundations of Structural Theory of Set Addition, volume 37 of Translations of Mathematical Monographs. AMS, 1973. http://www.emis.de/cgi-bin/zmen/ZMATH/en/quick.html?type=html&an=0271.10044Zbl 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 http://math.haifa.ac.il/ seva/pub_list.htmlhttp://math.haifa.ac.il/ seva/.
• 3 Melvyn B. Nathanson. Additive Number Theory: Inverse Problems and Geometry of Sumsets, volume 165 of GTM. Springer, 1996. http://www.emis.de/cgi-bin/zmen/ZMATH/en/quick.html?type=html&an=0859.11003Zbl 0859.11003.
Title Freiman isomorphism FreimanIsomorphism 2013-03-22 13:40:39 2013-03-22 13:40:39 bbukh (348) bbukh (348) 8 bbukh (348) Definition msc 11B75 msc 20K30 Isomorphism2