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Let $A$ and $B$ be subsets of abelian groups. A Freiman $2$-homomorphism is a function from $\phi:A\to B$ such that if $a_1 + a_2 = b_1 + b_2$, then $\phi(a_1)+\phi(a_2) = \phi(b_1) + \phi(b_2)$. If $\phi$ has an inverse that is also a Freiman $2$-homomorphism, then $\phi$ is said to be a $2$-isomorphism.

Now say $A$ contains $0$. Must $\phi(0) =0$ when $\phi$ is a $2$-isomorphism? My intuition says yes, but I have been unable to prove it. (We can't just say that since $0+a = a$, then $\phi(0) + \phi(a) = \phi(a)$, since we need another term on the right-hand side to apply the definition of $2$-homomorphism.)

If it is true, then does $\phi$ need to be a $2$-isomorphism? Or does $2$-homomorphism suffice? Thanks for the help!

marcelgoh
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  • What about $\phi:\mathbb{Z}/2\rightarrow\mathbb{Z}/2$ given by $\phi(0)=1$ and $\phi(1)=0$? – Christian E. Ramirez May 11 '23 at 00:08
  • @C-RAM ahh, that's interesting! I didn't expect an example like that. Can we say something weaker, maybe that $0$ is at least in the image of $A$ though? – marcelgoh May 11 '23 at 03:55
  • That still doesn't work. Let $a,b$ be any elements of a group $G$ where $b\ne 0$. The unique bijection $\phi:{a}\rightarrow {b}$ will satisfy the conditions. You're going to have to think more deeply about why these counterexamples exist if you want to salvage some weaker statement of this form (if it even exists). – Christian E. Ramirez May 11 '23 at 04:33
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    @C-RAM Ah, thanks. Sorry I asked this question as I was trying to understand a step in a proof that said "Freiman isomorphisms send subspaces of ${\bf F}_p^n$ to subspaces". But I missed in that statement that the first "subspace" means linear subspace and the second "subspace" means affine subspace. So of course an affine subspace need not contain $0$. If you would like to copy your first comment into an answer, I'd be happy to accept it. – marcelgoh May 11 '23 at 18:15
  • @marcelgoh, I am wondering, did this question arise from Gowers' lecture notes when he proves Freiman's theorem for subsets of $\mathbb{F}_p^N$? – RFZ Oct 02 '23 at 01:19

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