Rotation
Cyclic groups can represent finite rotations of symmetrical shapes in a plane.
c2
c3
Modulo Addition
Cyclic groups are equivalent to (isomorphic to) modulo n addition (denoted Zn). If we take the group operation to be + then we would tend to refer to Zn but if we take the group operation to be * then we would tend to refer to Cn.
Generators
If the group operation is multiplication then:
<r | r n =1>
If the group operation is addition then:
<z | n z = 0 >
Properties
Cyclic groups are Abelian which means that the group operation is commutative.
Cyclic Groups
A group whose elements can be written as e, a, a²… an-1
shifting rows
One possibility would be to start with a row containing all the elements in order, this is the 'identity' row:
| 0 | 1 | 2 | 3 |
Then shift the row to the right (modulo n).
| 1 | 2 | 3 | 0 |
Repeat this until we have done a complete cycle, then put all the rows above each other, the completed table is:
| 0 | 1 | 2 | 3 |
| 1 | 2 | 3 | 0 |
| 2 | 3 | 0 | 1 |
| 3 | 0 | 1 | 2 |
Generating a Cyclic Group using a Program
We can use a computer program to generate these groups, here I have used Axiom/FriCAS which is described here.
c1 := cyclicGroup(1)
<1>
Type: PermutationGroup(Integer)
toTable()$toFiniteGroup(c1,1)
Type: List(Matrix(Integer)) c2 := cyclicGroup(2)
<
(
1 2
)
>
Type: List(Matrix(Integer)) c3 := cyclicGroup(3)
<
(
1 2 3
)
>
Type: List(Matrix(Integer)) c4 := cyclicGroup(4)
<
(
1 2 3 4
)
>
permutationRepresentation(c4,4)
Type: List(Matrix(Integer)) c5 := cyclicGroup(5)
<
(
1 2 3 4 5
)
>
permutationRepresentation(c5,5)
|
where:
- The points of the permutation are numbered 1..n
- The elements of the group are named: "i" for the identity, single letters "a","b"... for the generators, and products of these.
- numbers in brackets are points of permutations represented in cyclic notation.
- cyclicGroup[1] is not really valid and the results for this case are nonsense.
- The Axiom/FriCAS program can't work in terms of the Cayley table, so I have added my own code to do this.
