Sidon subset of abelian group

This article defines a property of subsets of abelian groups
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Definition

A subset $S$ of an Abelian group $G$ is termed a Sidon subset if the equation:

$a+b=c+d$

has no solution for $a,b,c,d\in S$ other than where $a=b=c=d$.

(Note: In some variants, we define a Sidon subset as a subset where the above equation has no solutions for distinct $a,b,c,d$. This is a somewhat different definition, and allows for somewhat larger Sidon subsets.

Metaproperties

If $S$ is a Sidon subset of an Abelian group $G$, and $a\in G$, then $a+S$ is also a Sidon subset of $G$.

Any subset of a Sidon subset is also a Sidon subset.

Facts

• Any maximal Sidon subset cannot be contained in a proper subgroup, unless the quotient group has exponent two (Note: With the alternative definition, no maximal Sidon subset can be contained in a proper subgroup).