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).