Groups: Binary operations

Understanding Binary Operations in Groups

A group is mathematical structure consisting of a set of elements combined with an operation.
The operation combining two elements in the group is known as a binary operation.
This binary operation must follow a set of strict criteria to create a group.
These criteria include closure, associativity, an identity element, and the inverse element.

For closure, if two elements a and b belong to a group G, then the result of their binary operation, typically denoted as a*b or ab, must also be within G.
Associativity means that for three elements in the group G, a, b, and c, the operation should satisfy the following: (ab)c = a(bc).
A group must contain an identity element, often denoted by letter ‘e’ or the number ‘1’. The identity element is a special number such that when any element a in the group G is combined with this identity element, the original number remains unchanged, i.e., ae = ea = a.
Every element ‘a’ within the group G should have an inverse element, often denoted by a^(-1) or -a within G. When an element and its inverse are combined using the binary operation, the result is the identity element, i.e., aa^(-1) = a^(-1)a = e.

A subgroup is a smaller group contained inside a larger group. It is also a set of elements combined with the same binary operation.
A subgroup must satisfy the same group axioms: closure, associativity, identity, and inverse.

Group theory, including binary operations, has wide applications in different fields such as quantum physics, chemistry, and cryptography.
In quantum physics, group theory is used to develop quantum mechanics.
Chemists use group theory to understand the bonding and behavior of molecules.
In cryptography, group theory is used to create complex codes and decipher encrypted messages.

Familiar operations like addition (+) and multiplication (x) on the set of integers are examples of binary operations forming a group.
Matrix multiplication is a binary operation that can form a group when combined with the set of invertible matrices.
Function composition is a binary operation that forms a group with the set of bijections (one-to-one correspondences) from a set to itself.
Union and intersection of sets can be seen as binary operations and form a group when combined with the set of all subsets of a given set (forming the power set).