On a group of normalized solutions of the higher-dimensional Prouhet--Tarry--Escott problem

We elucidate, for the first time, a novel group-theoretic structure that arises from certain solutions of the $n$-dimensional Prouhet–Tarry–Escott problem of degree $2$ and size $n$. We prove that the group is isomorphic to the orthogonal group for a certain quadratic form.