Groups (Zn, U(n)) A group is a non-empty set G together with an operation (multiplication or addition) that combines any two elements a and b in G to for...
A group is a non-empty set G together with an operation (multiplication or addition) that combines any two elements a and b in G to form the element ab in G.
The set Zn, consisting of all integers under the operation of addition, is an example of a group. The operation is defined by the addition of two integers, and the identity element is 0, which is the sum of any two integers.
Additionally, the set U(n) of all positive integers under the operation of multiplication is an example of a group. The operation is defined by the multiplication of two integers, and the identity element is 1, which is the product of any two positive integers.
These examples demonstrate the vast diversity of groups and their wide applications in mathematics and other disciplines