Sidon subset of abelian group

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).