Freiman Homomorphisms Revisited

Abstract

Recall that if A ₁,…,A _n ⊆G are nonempty subsets of an abelian group G translated so that $0\in\bigcap_{i=1}^{n}A_{i}$, then a normalized Freiman homomorphism of $\sum _{i=1}^{n}A_{i}$ is a map $\psi :\sum _{i=1}^{n}A_{i}\rightarrow G'$, where G′ is another abelian group, such that

$$\psi(0)=0\quad\;\mbox{ and } \;\quad\psi\Biggl(\sum _{i=1}^{n}a_i\Biggr)=\sum _{i=1}^{n}\psi (a_i)\quad\mbox{ for } a_i\in A_i. $$

We will sometimes abbreviate the fact that the sumsets $\sum _{i=1}^{n}A_{i}$ and $\sum _{i=1}^{n}A'_{i}$ are Freiman isomorphic by writing

$$\sum _{i=1}^{n}A_i\cong \sum _{i=1}^{n}A'_i. $$

In such case, there exists a _i ∈A _i and $a'_{i}\in A'_{i}$ and a normalized Freiman isomorphism $\psi:\sum _{i=1}^{n}(-a_{i}+A_{i})\rightarrow \sum _{i=1}^{n}(-a'_{i}+A'_{i})$.

The goal of this chapter is to develop a basic theory of Freiman homomorphisms by introducing an algebraic invariant, the Universal Ambient Group (UAG), for a sumset $\sum _{i=1}^{n} A_{i}$. Among other things, this then allows us to show that sufficiently small subsets of an arbitrary abelian group have their sumset isomorphic to an integer sumset, which should be compared with previous results showing that any finite torsion-free sumset is isomorphic to an integer sumset. We will also show that any torsion-free sumset has a compact representation, derive several short exact sequences, derive an upper bound for the torsion subgroup of the UAG, and calculate the UAG under various small sumset hypotheses.