Graph Power of a Group

The graph power of a group by a finite simple graph with vertex set is the subgroup of the group direct product generated by closed-neighborhood activations. For and , let be the configuration equal to on the closed neighborhood and to the identity element elsewhere. Then

Right multiplication by multiplies the states in on the right by . Thus is the collection of configurations reachable from the all-identity configuration in a group-valued Lights Out puzzle (Cunningham and Minevich 2026, p. 5).

For the cyclic group of group order 2, this recovers the usual binary game. More generally, for a cyclic group of group order , reachability is a linear congruence equation problem with activation matrix , where is the identity matrix (Edgar and Sklar 2016; Cunningham and Minevich 2026, pp. 3-5). For a complete graph , only constant configurations are reachable, so is the diagonal copy of . For the three-vertex path graph , (Cunningham and Minevich 2026, p. 6).

A graph is called reducible to abelian when for every group , where is the commutator subgroup. In this case, reachability is determined by the corresponding problem over the abelianization . Let be the integer matrix whose rows are the indicator functions of , with all vertices allowed, including . Then is reducible to abelian iff the rows generate the entire integer lattice:

Equivalently, has matrix rank and every nonzero diagonal entry of its Smith normal form is 1 (Cunningham and Minevich 2026, pp. 13, 20, 24).