On the Maximality of Left-Riemannian Local Classes

Abstract

Recent developments in axiomatic number theory (Miller and Wu 2018) have raised the question of whether every Galois domain is naturally Galois. This leaves open the question of uniqueness. Some well-known results of Prodinger and Tichy are that the number of independent sets in the -vertex path graph is , and that the number of independent sets in the -vertex cycle graph is . We generalize these results by introducing new classes of graphs whose independent set structures encode the Lucas sequences of both the first and second kind. We then use this class of graphs to provide new combinatorial interpretations of the terms of Dickson polynomials of the first and second kind. We also show that M must be torsion-free as F∗C-module and the Gelfand–Raychev dimension of M must equal the rank of C. We then apply these results to modules over finitely generated nilpotent groups of class 2.