Maths - Cyclic Groups

Rotation

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

c2

c2 graph

c3

c3 graph

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)
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:


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