Subtle Internal Structural Symmetries Around Group ℤn

The main theme of the paper is around the claim, “The structure of group ℤ12 and ℤn in general, can be described as the union between a set and its own rotation and reflection. This set is a subset of ℤn. The process is commutative and reversible.” Before substantiating the claim it is shown that ℤ, the set of integers can be shown as the union of a set ℕ and its reflection around zero, r0. The claim is then substantiated. Having substantiated the claim, the paper returns to the number line and shows depending on how we take natural numbers and zero, the process is either reversible or not. When zero is its own set and not part of the natural numbers the process is reversible. Extensive use of tables is used such that one can see for themselves the process. At the end a new class of groups is presented.