A group is a finite or infinite set of elements together with a binary operation (called the group operation) that together satisfy the four fundamental properties of closure, associativity, the identity property, and the inverse property. The operation with respect to which a group is defined is often called the "group operation," and a set is said to be a group "under" this operation. Elements , , , . with binary operation between and denoted form a group if 1. Closure: If and are two elements in , then the product is also in . 2. Associativity: The defined multiplication is associative, i.e., for all , . 3. Identity: There is an identity element (a.k.a. 1, , or ) such that for every element . 4. Inverse: There must be an inverse (a.k.a. reciprocal) of each element.
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