Cyclic group:Z6

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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 $x^l{x}^m=x^{l+m}$ 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.