The main goal in this section is to prove the Fundamental Theorem of Abelian Groups, which roughly speaking says that every finite, abelian group is isomorphic to a product of cyclic groups. In other words, if A is a finite, abelian group, then there are positive integers, n1, . . . , nr, so that A ∼= Z/n1Z× · · · × Z/nrZ. It’s possible to have Z/n1Z × · · · × Z/nrZ ∼= Z/m1Z × · · · × Z/msZ whe...

In this paper we show how to construct a Dirac operator on a lattice in complete analogy with the continuum. In fact we consider a more general problem, that is, the Dirac operator over an abelian finite group (for which a lattice is a particular example). Our results appear to be in direct connexion with the so called fermion doubling problem. In order to find this Dirac operator we need to in...

We investigate the descriptive complexity of finite abelian groups. Using Ehrenfeucht-Fräıssé games we find upper and lower bounds on quantifier depth, quantifier alternations, and number of variables of a first-order sentence that distinguishes two finite abelian groups. Our main results are the following. Let G1 and G2 be a pair of non-isomorphic finite abelian groups, and let m be a number t...

We consider the question of when a random walk on a finite abelian group with a given step distribution can be used to reconstruct a binary labeling of the elements of the group, up to a shift. Matzinger and Lember (2006) give a sufficient condition for reconstructibility on cycles. While, as we show, this condition is not in general necessary, our main result is that it is necessary when the l...