# Order dividing bijective function from non-cyclic to cyclic groups of same finite order

## TL;DR

This paper constructs explicit bijective functions that map between cyclic and non-cyclic finite groups, demonstrating order-dividing correspondences for specific group classes.

## Contribution

It introduces new order-dividing bijections between certain non-cyclic and cyclic finite groups, expanding understanding of their structural relationships.

## Key findings

- Existence of a bijection from D_{2n} to Z_{2n} for all natural n.
- Construction of a bijection from Z_p x Z_k to Z_{pk} when p is an odd prime and p does not divide k.
- Demonstrates order-dividing properties in these group mappings.

## Abstract

In this article we give an order-dividing bijective function between cyclic and non cyclic groups of finite order. In particular, we prove that there exists a bijective function from D_{2n} to Z_{2n} for any natural integer n; and from Z_p x Z_k to Z_{pk} when p is an odd prime and k is not a multiple of p.