# 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 | aG

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

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}

Where I can, I have put links to Amazon for books that are relevant to the subject, click on the appropriate country flag to get more details of the book or to buy it from them.

 The Princeton Companion to Mathematics - This is a big book that attempts to give a wide overview of the whole of mathematics, inevitably there are many things missing, but it gives a good insight into the history, concepts, branches, theorems and wider perspective of mathematics. It is well written and, if you are interested in maths, this is the type of book where you can open a page at random and find something interesting to read. To some extent it can be used as a reference book, although it doesn't have tables of formula for trig functions and so on, but where it is most useful is when you want to read about various topics to find out which topics are interesting and relevant to you.