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# 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 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. Zbl 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/.
- 3 Melvyn B. Nathanson. Additive Number Theory: Inverse Problems and Geometry of Sumsets, volume 165 of GTM. Springer, 1996. Zbl 0859.11003.

## Mathematics Subject Classification

11B75*no label found*20K30

*no label found*

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