Proof that Every Cyclic Group is Abelian

Every cyclic group is Abelian. Prove it.

Cyclic groups and Abelian groups are fundamental structures in the realm of abstract algebra. Understanding why every cyclic group is Abelian requires a grasp of the definitions and intrinsic properties of these groups. A cyclic group is one that is generated by a single element, meaning every element in the group can be expressed as a power (or multiple, in additive terms) of that generator. On the other hand, an Abelian group is characterized by its commutative property; for any two elements in the group, the result of their operation is independent of their order.

The proof that every cyclic group is Abelian revolves around these fundamental properties. When you consider a cyclic group generated by an element 'a', any two elements in the group can be written as powers of 'a', say 'a^m' and 'a^n'. Using the property of exponents and the definition of the group operation, we find that the order of operation does not affect the outcome. This inherently demonstrates the commutativity of the group operation for cyclic groups.

This property is pivotal in understanding the broader algebraic structures, as it connects to how groups are classified and the types of operations that can be performed within them. Specifically, cyclic groups serve as building blocks for many other algebraic structures and concepts, providing a basis from which further algebraic theories can be developed, such as those involving homomorphisms and isomorphisms, and applications of Lagrange's Theorem.