Maths - Factoring Groups

On the previous pages we saw how to combine two or more groups into a bigger group, here we look at the reverse process, that is, breaking a group down into two or more simpler groups.

Lets assume that we have a group G and we have a subgroup of G which we call N. We want to find the other subgroup G/N. For G/N to be a group then N must be a normal subgroup.

Test that N is a normal subgroup

To test that N is a normal subgroup then:

x N x-1 must be a member of N for every x in G

Calculation of G/N

we calculate the set:

aN | a∈G

This will be the group G/N under the operation

(a N) (b N) = a b N

Example 1 - divide C by Z2

G = Complex numbers = C

N = Z2

Test for Normal Subgroup

x x N x-1
1 1 {1,-1} 1 = {1,-1}
-1 -1 {1,-1} -1 = {1,-1}
i i {1,-1} -i = {1,-1}
-i -i {1,-1} i = {1,-1}

So x N x-1 is a member of N for every x which means that G/N will be a group and we can go on to calculate it.

Calculation of G/N

Elements of G/N are a•N:

1•{1,-1} = {1,-1}
-1•{1,-1} = {-1,1}
i•{1,-1} = {i,-i}
-i•{1,-1} = {-i,i}

so the elements are:

{1,-1} and {i,-i}

Example 2 divide H by C

G is the group of quaternions H:

Cayley Table
Cayley Graph
a*b b.1 b.i b.j b.k
a.1 1 i j k
a.i i -1 k -j
a.j j -k -1 i
a.k k j -i -1
quaternion cayley digraph

Note: in this example I have not shown the negative elements so where i is shown we also have -i and so on for the other elements.

We want to divide it by C to get H/C

Test for Normal Subgroup

x x K x-1
1 1 {1,i} 1 = {1,i}
i i {1,i} -i = {-1,-i}
j j {1,i} -j = {-1,-i}
k k {1,i} -k = {-1,-i}

So x K x-1 is a member of K for every x which means that G/K will be a group and we can go on to calculate it.

Calculation of G/N

Elements of G/N are a•N:

1•{1,i} = {1,i}
i•{1,i} = {-1,i}
j•{1,i} = {j,-k}
k•{1,i} = {k,-j}

So the elements are:

{±1,±i} and {±j,±k}

which gives the multipication table:

a*b {±1,±i} {±j,±k}
{±1,±i} {±1,±i} {±j,±k}
{±j,±k} {±j,±k} {±1,±i}


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