How does an Abelian group differ from a noncommutative group?

In modern algebra, the primary distinction between an Abelian group and a noncommutative group lies in whether the group satisfies the commutative law.1

• Abelian Group: Also known as a commutative group, an Abelian group is one in which the order of the elements does not affect the result of the binary operation. For any two elements $a$ and $b$ in the group, the operation must satisfy $a*b=b*a$. A common example is the set of integers under addition.12
• Noncommutative Group: A group is noncommutative if there exists at least one pair of elements $a$ and $b$ such that $a*b\ne b*a$. In these groups, the order in which elements are combined is significant. A classic example is the set of rotations of a three-dimensional object around a fixed point, where performing rotations in a different sequence results in a different final orientation.2

Both types of groups must satisfy the fundamental group axioms: closure, the associative law, the existence of an identity element, and the existence of an inverse for every element.12