Groups: Cyclic groups

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.