On structures induced by the power sequences of $($\mathbb{Z}/ m\mathbb{Z}$, \cdot)$

In this paper, we explore the structure of $\mathbb{Z}/ m\mathbb{Z}$ in terms of its orbits under modular exponentiation, illustrating this with a sequential power graph that is naturally derived from the orbits by connecting elements of $\mathbb{Z}/ m\mathbb{Z}$ in the orbit order in which they appear. We find that this graph has a great deal of fascinating algebraic structure. The connected components are composed of orbits that all share at least one element. The vertex sets of the connected components are shown to depend on the factorization of $m$; in fact, the connected components are completely determined by the units of $\mathbb{Z}/ m\mathbb{Z}$, the idempotents of $\mathbb{Z}/ m\mathbb{Z}$ and the square-free divisors of $m$. Both tails and non-tails of the components can be described explicitly and algebraically in terms of these elements of $\mathbb{Z}/ m\mathbb{Z}$. Finally, a lattice of components can be used to show homomorphisms between the non-tails of any two comparable components in the lattice. This extensive structure is used here to prove an algebraic identity on the roots of an idempotent mod $m$, and may be exploited to prove other identities as well.