Maths - Normal Subgroups

If N is a subgroup of G then N is a normal subgroup if:

g N = N g

for all g in G

This does not necessarily mean that g n = n g for all g∈G, n∈N but it means that the set {g N} is the same as the set {N g} or, in other words,

g n = n' g

That is, if we choose an n in N then n' will also be a member of N (although not necessarily the same member).

If we multiply the above equation by g-1 we get:

n' = g n g-1


This second form is probably the best way to test if N is a normal subgroup.

Calculation of G/N

we calculate the set:

g N | g∈G

This will be the group G/N under the operation

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

Example 1

Lets take the example of C3×C2 :

generator cayley graph table
<m,r | m²,r³,rm=mr> c3c2 graph
1 r m rm r²m
r 1 rm r²m m
1 r r²m m rm
m rm r²m 1 r
rm r²m m r 1
r²m m rm 1 r

In this example we will test if C3 is a normal subgroup and then divide by it to check that the result is C2.

First we need to know the inverse function, to do this we take a particular column, go down until we get to the identity element '1' and then read out the row:

x x-1
1 1
m m
rm r²m
r²m rm

let R = the elements of C3, that is, 1 , r and r²:

x x 1 x-1 x r x-1 x r² x-1
1 1 1 1 = 1 1 r 1 = r 1 r² 1 = r
r r² 1 r² = r r² r r² = r² r² r² r² = 1
r 1 r = r² r r r = 1 r r² r = r
m m 1 m = 1 m r m = r m r² m = r²
rm r²m 1 rm = 1 r²m r rm = r r²m r² rm = r²
r²m rm 1 r²m = 1 rm r r²m = r rm r² r²m = r²

So the result is always a member of C3 so it is a normal subgroup. We can now try divideing by C3 as follows:

The result of this division has elements R and Rm:

R Rm
Rm R

Example 2 - 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 3 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}

metadata block
see also:


Correspondence about this page

Book Shop - Further reading.

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.

flag flag flag flag flag flag 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.


Terminology and Notation

Specific to this page here:


This site may have errors. Don't use for critical systems.

Copyright (c) 1998-2023 Martin John Baker - All rights reserved - privacy policy.