Cyclic group:Z6

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Definition

This group, denoted $C_{6},\mathbb {Z} _{6}$ or $\mathbb {Z} /6\mathbb {Z}$, is defined in the following equivalent ways:

1. It is a cyclic group of order $6$.
2. It is the direct product of the cyclic group of order three and the cyclic group of order two.

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.

Properties

Property	Satisfied?	Explanation	Comment
Abelian group	Yes	Cyclic implies abelian	It is the only abelian group of order 6. The only other group of order 6 is dihedral group:D6, which is non-abelian.
Nilpotent group	Yes	Abelian implies nilpotent
Leinster group	Yes	Cyclic group of perfect number order is Leinster group

GAP implementation

Group ID

This finite group has order 6 and has ID 2 among the groups of order 6 in GAP's SmallGroup library. For context, there are groups of order 6. It can thus be defined using GAP's SmallGroup function as:

SmallGroup(6,2)

For instance, we can use the following assignment in GAP to create the group and name it $G$:

gap> G := SmallGroup(6,2);

Conversely, to check whether a given group $G$ is in fact the group we want, we can use GAP's IdGroup function:

IdGroup(G) = [6,2]

or just do:

IdGroup(G)

to have GAP output the group ID, that we can then compare to what we want.