One promising technique for understanding features of nonlocal games is to study constraints placed on the players' measurement operators using techniques from algebraic combinatorics. In this talk, I will show an XOR game has commuting operator value 1 iff an instance of the subgroup membership problem on a finitely presented group corresponding to the game has a solution. This relationship can be used to show that the value one question is decidable for interesting sub-cases of XOR games. It also gives an algebraic framing of some open questions concerning XOR games. Based on joint work with Aram Harrow, Anand Natarajan, and Gurtej Kanwar.