Which rules must a set and operation satisfy to form a group?

In modern algebra, a group is defined as a set of elements combined with a binary operation that must satisfy four fundamental axioms:1

• Closure: The set must be closed under the operation, meaning that combining any two elements of the set always produces another element within the same set.1
• Associativity: The operation must obey the associative law, where for any elements $a,b,$ and $c$, the equation $a(bc)=(ab)c$ holds true.12
• Identity Element: The set must contain an identity element that, when combined with any other element in the set, leaves that element unchanged.1
• Inverse Element: Each element in the set must have an inverse element such that their combination results in the identity element.12

If a group also satisfies the commutative law, it is classified as a commutative, or abelian, group.1