Cyclic group:Z243
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Contents
Definition
This group is defined as the cyclic group of order , and is denoted
,
or
.
Arithmetic functions
Function | Value | Explanation |
---|---|---|
order | 243 | |
prime-base logarithm of order | 5 | |
exponent | 243 | |
prime-base logarithm of exponent | 5 | |
minimum size of generating set | 1 | |
subgroup rank | 1 | |
rank as p-group | 1 | |
normal rank | 1 | |
characteristic rank | 1 | |
derived length | 1 | |
nilpotency class | 1 | |
Frattini length | 5 |
GAP implementation
Group ID
This finite group has order 243 and has ID 1 among the groups of order 243 in GAP's SmallGroup library. For context, there are groups of order 243. It can thus be defined using GAP's SmallGroup function as:
SmallGroup(243,1)
For instance, we can use the following assignment in GAP to create the group and name it :
gap> G := SmallGroup(243,1);
Conversely, to check whether a given group is in fact the group we want, we can use GAP's IdGroup function:
IdGroup(G) = [243,1]
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 as:
CyclicGroup(243)
or
CyclicGroup(3^5)