# Groups: Binary operations

## 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**.

## Characteristics of Binary Operations

- 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: (a*b)*c = a*(b*c).- 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., a*e = e*a = 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., a*a^(-1) = a^(-1)*a = e.

## Subgroups and Binary Operations

- 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.

## Applications of Binary Operations and Group Theory

- 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.

## Examples of Binary Operations in Groups

- 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).