# Maths - Cyclic Groups

### Rotation

Cyclic groups can represent finite rotations of symmetrical shapes in a plane.

### c2 ### c3 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)```
 a a a a
Type: Table(2) permutationRepresentation(c1,1)
[
 1
]

Type: List(Matrix(Integer))

c2 := cyclicGroup(2) < ( 1 2 ) >
Type: PermutationGroup(Integer)
toTable()\$toFiniteGroup(c2,1)

 i a a i
Type: Table(2) permutationRepresentation(c2,2)
[
 0 1 1 0
]

Type: List(Matrix(Integer))

c3 := cyclicGroup(3) < ( 1 2 3 ) >
Type: PermutationGroup(Integer)
toTable()\$toFiniteGroup(c3,1)

 i a aa a aa i aa i a
Type: Table(3) permutationRepresentation(c3,3)
[
 0 0 1 1 0 0 0 1 0
]

Type: List(Matrix(Integer))

c4 := cyclicGroup(4) < ( 1 2 3 4 ) >
Type: PermutationGroup(Integer)
toTable()\$toFiniteGroup(c4,1)

 i a aa aaa a aa aaa i aa aaa i a aaa i a aa
Type: Table(4)
permutationRepresentation(c4,4)
[
 0 0 0 1 1 0 0 0 0 1 0 0 0 0 1 0
]

Type: List(Matrix(Integer))

c5 := cyclicGroup(5) < ( 1 2 3 4 5 ) >
Type: PermutationGroup(Integer)
toTable()\$toFiniteGroup(c5,1)

 i a aa aaa aaaa a aa aaa aaaa i aa aaa aaaa i a aaa aaaa i a aa aaaa i a aa aaa
Type: Table(5)
permutationRepresentation(c5,5)
[
 0 0 0 0 1 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0
]
Type: List(Matrix(Integer)) (16) ->

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 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.      Symmetry and the Monster - This is a popular science type book which traces the history leading up to the discovery of the largest symmetry groups.