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Abstract
Group theory studies the algebraic structure known as groups. A group is a non-empty set with a binary operation that satisfies some axioms. These axioms include associativity, the existence of an identity element, and the existence of an inverse element. An Abelian group is a group that satisfies the commutative law. One of the most well-known examples of a group is the cyclic group , whose elements repeat periodically after steps under modular addition. On the other hand, planetary rotation and revolution are continuous dynamical processes that repeat periodically and are governed by gravitational mechanics. In this research, we construct a discrete mathematical model of planetary periodic motion. Since physical planetary motion is continuous, we discretize each planet's rotation and revolution period to the nearest positive integer in an appropriate time unit, yielding a finite set of positional states , where represents the planet's position at time step . We define the binary operation , which represents the composition of two successive time progressions modulo the period. We then verify that satisfies the group axioms, forming an Abelian group isomorphic to . Our purpose here is to show some applications of algebra outside the classroom.
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