Cyclic group:Z6

From Groupprops
Jump to: navigation, search
This article is about a particular group, i.e., a group unique upto isomorphism. View specific information (such as linear representation theory, subgroup structure) about this group
View a complete list of particular groups (this is a very huge list!)[SHOW MORE]

Definition

Verbal definition

The cyclic group of order 6 is defined as the group of order six generated by a single element. Equivalently it can be described as a group with six elements e= x^0,x,x^2,x^3,x^4,x^5 where x^lx^m = x^{l+m} with the exponent reduced mod 3. It can also be viewed as:

  • The quotient group of the group of integers by the subgroup of multiples of 6.
  • The multiplicative group comprising the six sixth roots of unity (as a subgroup of the multiplicative group of nonzero complex numbers)
  • The group of orientation-preserving symmetries (rotational symmetries) of the regular hexagon.