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. Therefore, for each element 
of 
, the set contains an element 
such that 
.
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