How do you show abelian?

Excuse me, but could you please elaborate on how one might demonstrate the properties of an abelian group in a clear and concise manner? I'm particularly interested in understanding the key elements and processes involved in illustrating this concept, as it's a fundamental aspect of algebra and cryptography that I'm currently studying. Would you be able to guide me through an example or provide a step-by-step approach to demonstrating abelian properties?