Creat fp group using addition relation.

[Hongyi Zhao]

As commented here [1], the following two methods  can be used to define an Cyclic Group:

```
Generators
If the group operation is multiplication then:
<r | r ^ n =1>
If the group operation is addition then:
<z | n * z = 0 >
```

For the first case, the corresponding code snippet in GAP is as follows:

```

gap> f:=FreeGroup("P");;
gap> g:=f/ParseRelators(f, "P^8" );
<fp group on the generators [ P ]>
gap> StructureDescription(g);
"C8"
```

But I'm not sure if GAP also supports the second method mentioned above to define a group.

[1] http://www.euclideanspace.com/maths/discrete/groups/categorise/types/abelian/cyclic/index.htm

Regards,

HZ

[David Joyner]

On Wed, May 25, 2022 at 6:08 AM Hongyi Zhao <hongy...@gmail.com> wrote:
>
> As commented here [1], the following two methods can be used to define an Cyclic Group:
>
> ```
> Generators
> If the group operation is multiplication then:
> <r | r ^ n =1>
> If the group operation is addition then:
> <z | n * z = 0 >
> ```
>
> For the first case, the corresponding code snippet in GAP is as follows:
>
> ```
> gap> f:=FreeGroup("P");;
> gap> g:=f/ParseRelators(f, "P^8" );
> <fp group on the generators [ P ]>
> gap> StructureDescription(g);
> "C8"
> ```
>
> But I'm not sure if GAP also supports the second method mentioned above to define a group.
>

While SageMath does use GAP for a lot of group theory, it uses different command
syntax for cyclic groups. In SageMath, you can use both
CyclicPermutationGroup(8)
for the multiplicative cyclic group of order 8, or
IntegerModRing(8)
for the additive version. Please see the additional examples given in
the tutorials
https://doc.sagemath.org/html/en/thematic_tutorials/group_theory.html
and
https://doc.sagemath.org/html/en/constructions/groups.html

> [1] http://www.euclideanspace.com/maths/discrete/groups/categorise/types/abelian/cyclic/index.htm
>
> Regards,
> HZ
>

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[Hongyi Zhao]

On Wednesday, May 25, 2022 at 6:22:48 PM UTC+8 wdjo...@gmail.com wrote:

On Wed, May 25, 2022 at 6:08 AM Hongyi Zhao <hongy...@gmail.com> wrote:
>
> As commented here [1], the following two methods can be used to define an Cyclic Group:
>
> ```
> Generators
> If the group operation is multiplication then:
> <r | r ^ n =1>
> If the group operation is addition then:
> <z | n * z = 0 >
> ```
>
> For the first case, the corresponding code snippet in GAP is as follows:
>
> ```
> gap> f:=FreeGroup("P");;
> gap> g:=f/ParseRelators(f, "P^8" );
> <fp group on the generators [ P ]>
> gap> StructureDescription(g);
> "C8"
> ```
>
> But I'm not sure if GAP also supports the second method mentioned above to define a group.
>

While SageMath does use GAP for a lot of group theory, it uses different command
syntax for cyclic groups. In SageMath, you can use both
CyclicPermutationGroup(8)
for the multiplicative cyclic group of order 8, or
IntegerModRing(8)
for the additive version.

Thank you for your tips and tricks. In fact, I'm working in SageMath with GAP, so I'm interested in the additive version implemented in GAP, and am not sure if it also such a counterpart.

Best,

HZ