Started learning Group Theory, and learned about the integers modulo n. Decided to code up a visualization of the additive operation generators on the set of elements in the group of integers modulo n. This generator must have no prime factors in common with n otherwise it would not reach every number in the set.
This code creates a manipulate for

modn and gives each possible generator that reaches the whole set. The elements are shown on the points of a n-sided polygon.
In[]:=
intmodn[modulo_Integer]:=Manipulate[Show[Graphics[{LightBlue,RegularPolygon[modulo]},PlotLabel->StringForm["<``,+``>",modulo,op]],Table[Graphics[Text[Style[n,20,Red],CirclePoints[modulo][[n+1]]]],{n,0,modulo-1}],Graphics[Circle[CirclePoints[modulo][[Mod[op*i,modulo]+1]],.1]]],{i,1,modulo,1},{op,Select[Range[0,modulo-1],GCD[#,modulo]==1&]}]

Integers Modulo 12

In[]:=
intmodn[12]
Out[]=
​
i
op
1
5
7
11
We see the possible generators are 1, 5, 7, 11. This is because these are the only numbers in the set of

mod12 that have no prime factors in common with 12.
We see that 1 and 11 have an interesting property of mirroring their behavior. 1 goes counterclockwise from one to zero, 11 goes clockwise from 11 to one.

Integers Modulo 17

In[]:=
intmodn[17]
Out[]=
​
i
op
4
Since 17 is a prime number each of the elements in

mod12 can be an add operation for traversing the elements.

Integers Modulo 4

We see the integers module 4 with the addition operation has an isomorphism to the group of rotations of a square.
In[]:=
intmodn[4]
Out[]=
​
i
op
1
3
In[]:=
Table[Rotate[Show[Graphics[{LightBlue,RegularPolygon[4]}],Table[Graphics[Text[Style[n,20,Red],CirclePoints[4][[n+1]]]],{n,0,3}]],rot],{rot,{0,Pi/2,Pi,3*Pi/2}}]
Out[]=

,
,
,

This is also generalizable to all the other n-sided polygons.

CITE THIS NOTEBOOK

Visualization for the generators of the additive group of integers modulo n​
by Arihant Gadgade​
Wolfram Community, STAFF PICKS, October 13, 2024
​https://community.wolfram.com/groups/-/m/t/3297140