How to solve an abelian group?

How might one approach solving an abelian group, considering its defining characteristic of commutativity? What specific methods or techniques could be employed to analyze and understand the structure of such a group? Is there a systematic way to break down the problem into smaller, more manageable parts, or is it necessary to take a more holistic approach? What role does understanding the order and generators of the group play in finding a solution? And finally, are there any common pitfalls or misconceptions that one should be aware of when working with abelian groups?