Cyclic group:Z18
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Contents
Definition
This group, denoted ,
,
, is defined in the following equivalent ways:
- It is the direct product of cyclic group of order nine and cyclic group of order two.
- It is the cyclic group of order
.
Arithmetic functions
Function | Value | Explanation |
---|---|---|
order | 18 | |
exponent | 18 | |
Frattini length | 2 |
GAP implementation
Group ID
This finite group has order 18 and has ID 2 among the groups of order 18 in GAP's SmallGroup library. For context, there are groups of order 18. It can thus be defined using GAP's SmallGroup function as:
SmallGroup(18,2)
For instance, we can use the following assignment in GAP to create the group and name it :
gap> G := SmallGroup(18,2);
Conversely, to check whether a given group is in fact the group we want, we can use GAP's IdGroup function:
IdGroup(G) = [18,2]
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(18)