Cyclic group:Z30

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Definition

This group, denoted $C_{30},\mathbb {Z} _{30}$ or $\mathbb {Z} /30\mathbb {Z}$, is defined in the following equivalent ways:

1. It is a cyclic group of order $30$.
2. It is the direct product of cyclic group:Z2, cyclic group:Z3 and cyclic group:Z5.

Verbal definition

It can also be viewed as:

• The quotient group of the group of integers by the subgroup of multiples of 30.
• The group of orientation-preserving symmetries (rotational symmetries) of the regular 30-gon.

Properties

Property	Satisfied?	Explanation	Comment
Abelian group	Yes	Cyclic implies abelian
Nilpotent group	Yes	Abelian implies nilpotent

GAP implementation

Group ID

This finite group has order 30 and has ID 4 among the groups of order 30 in GAP's SmallGroup library. For context, there are groups of order 30. It can thus be defined using GAP's SmallGroup function as:

SmallGroup(30,4)

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

gap> G := SmallGroup(30,4);

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

IdGroup(G) = [30,4]

or just do:

IdGroup(G)

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

Other descriptions

The group can be defined using GAP's CyclicGroup function:

CyclicGroup(30)