The equation $|A^{-1}A|=|A|^2-|A|+1$ for finite subsets of a group

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Let $A$ be a finite subset of a group $G$. It is clear that $|A^{-1}A|\leq|A|^2-|A|+1$. If $A$ is singleton or $A=\{1,a\}$ with $O(a)\neq 2$ then the equality holds. Now, can somebody characterize all $A$ with the property in some important groups (finite or infinite)? $$|A^{-1}A|=|A|^2-|A|+1$$

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Such sets are called Golomb rulers, or equivalently Sidon sets. At least when $G$ is abelian, they have been extensively studied. I doubt there is a simpler characterization than your definition.