# Groups: Cyclic groups

## Understanding Groups and Cyclic Groups

• A group is a set of elements combined with a binary operation that satisfies certain properties including closure, associativity, identity and inverse.
• A cyclic group is a special type of group where every element in the group can be written as a power of one specific element.
• This specific element is known as a generator of the group, and it cycles through all the elements in the groups as it is raised to different powers.

## Properties of Cyclic Groups

• Each cyclic group has at least one generator, and multiple different generators may exists depending on the size of the group.
• Every subgroup within a cyclic group is also a cyclic group.
• If a cyclic group has ‘n’ elements, it is commonly denoted as Cn or Zn.
• All cyclic groups are Abelian groups, which means the group operation is commutative (i.e., the order in which elements are combined does not matter).

## Applications of Cyclic Groups

• Cyclic groups are applied in different areas of mathematics including number theory, geometry, and algebra.
• They also have essential applications in cryptography where concepts of cyclic groups are used in creating secure communication channels.
• Properties of cyclic groups aid in problem-solving especially in domain areas where periodicity, rotations, and recurrences are involved.

## Operations on Cyclic Groups

• A binary operation combines two elements to produce another element within the group.
• This operation adheres to the group properties, so for a given cyclic group ‘G’, an operation ‘*’ and elements ‘a’ and ‘b’ in ‘G’, the result of ‘a * b’ is also in ‘G’.
• The order of an element within a cyclic group is the smallest positive power to which the element must be raised to get the identity of the group.

## Finite and Infinite Cyclic Groups

• Cyclic groups can either be finite or infinite. In a finite cyclic group, the generator eventually repeats itself, while in an infinite cyclic group it doesn’t.
• Subgroups of finite cyclic groups can be identified by the factors of the order of the main group, and there will be exactly one subgroup for each divisor.
• Each element of an infinite cyclic group generates a unique subgroup.