# Generators

## Defining Generators in Groups

- In the context of group theory, a
**generator**is an element from which all other elements of a group can be obtained using only the group operation and the operation of taking inverses. - A group is said to be
**generated**by a set of elements if every element in the group can be expressed as the finite product of these elements and their inverses. - A group that can be generated by a single element is called a
**cyclic group**.

## Properties of Generators

- The concept of a generator is intrinsically related to the
**order**of a group. The order of an element ‘g’ in a group is the smallest positive integer ‘n’ such that g^n equals the identity element. - If for a group element ‘g’, there is a positive integer ‘n’ such that g^n equals the identity, ‘g’ is said to have finite order. If no such integer exists, ‘g’ is said to have infinite order.
- Any element of a group generates a
**subgroup**, the set of all powers of that element.

## Generators of Finite and Infinite Groups

- In both finite and infinite groups, the generators play a key role in determining the structure of the group.
- In
**finite groups**, the order of a generator divides the order of the group. This is known as Lagrange’s theorem. - For
**infinite cyclic groups**, each non-identity element generates a copy of the integers under addition. This implies that an infinite cyclic group has exactly two generators.

## Examples of Generators

- For the group of integers under addition,
**Z**, the generators are 1 and -1. - The group of integers mod n, denoted
**Z_n**, is a finite cyclic group where 1 is a generator. - The group of complex nth roots of unity is generated by e^(2πi/n).