It is a famous open problem to estimate non-trivially, for a prime $p\equiv 1\pmod 4$, the largest size of a subset $A\subset{\mathbb F}_p$ such that the difference of any two elements of $A$ is a square in ${\mathbb F}_p$; some basic information can be found [here][1]. The trivial estimate is $|A|<\sqrt p+O(1)$, which is easy to obtain in several ways. Can we do better if $A$ is known to be a subgroup of the multiplicative group of ${\mathbb F}_p$?

> For a prime $p\equiv 1\pmod 4$, how large can a subgroup $H<{\mathbb F}_p^\times$ be given that the difference of any two elements of $H$ is a square?

Equivalently, 

> For a prime $p\equiv 1\pmod 4$, denoting by $\mathcal Q$ the set of all squares in ${\mathbb F}_p$, what is the largest size of a subgroup of ${\mathbb F}_p^\times$ contained in ${\mathcal Q}\cap({\mathcal Q}+1)$?

Since containment in ${\mathcal Q}$ is not that much restrictive for a subgroup, this seems essentially equivalent to asking about the largest possible size of a subgroup contained in ${\mathcal Q}+1$.




[1]: http://mathoverflow.net/questions/48591/cliques-paley-graphs-and-quadratic-residues