Activity on Cosets, Normal Subgroups and Quotient Groups
This is an interactive web page containing applets. They should
load and run in any reasonably modern browser. Solutions to
questions can be viewed by clicking a button. You may, if you wish
download the pdf file of the
solutions.
Similarly you may,
if you wish, download the pdf file of the
activity.
These will need Adobe acrobat to read and print.
Proposed Learning Outcomes
This activity uses small finite groups to reinforce the idea of
cosets, normal subgroups and quotient groups. It will also revise
the idea of isomorphism. You will
 remind yourself about subgroups;
 see individual left and right cosets of various subgroups
of several small finite groups in the context of a Cayley table;
 see how a Cayley table can be rearranged so that the left
cosets are grouped together;
 see how a Cayley table can be rearranged so that the right
cosets are grouped together;
 be alerted to the difference between normal subgroups and
nonnormal subgroups.
 be able to investigate various quotient groups and visually
see isomorphisms between these and groups with which you are
familiar;
 be made aware that isomorphic normal subgroups may have
nonisomorphic quotient groups.
Subgroups and their cosets
The java applet on the left shows the Cayley tables of various
small finite groups. The table shown initially is the table for
S(^{[¯]}). This is the group of symmetries of a square, where
a,b,c are rotations through p/2,p and 3p/2
respectively and r,s,t,u are reflections. For each group, listed
in the `list box' marked `G', that you choose, a selection of
subgroups are listed in the `list box' marked `H'. These are
given, either as a list of elements or, when cyclic, in terms of a
generator. Thus áañ indicates the subgroup
generated by a. Which elements will belong to it? Click on
this subgroup now, and observe how the elements of the subgroup
are highlighted in the top row and left column.
Click on ábñ and see that the elements of this
subgroup are highlighted.
Activity 1
In this activity you will explore left
cosets of subgroups of
S(^{[¯]}) and of subgroups of at least
one other group.
Make sure you have the group
S(^{[¯]}) selected. We are going
to start by looking at left cosets of áañ. In
general the left coset gH of a subgroup H of a group
G, generated by g is defined as
For
the subgroup áañ in
S(^{[¯]}), this means
that the left coset for ráañ, say is the set
Can you see from the Cayley
table what the elements will be?
 Click on the subgroup áañ and then click on
the element r in the leftmost column of the Cayley table.
Observe how the elements of the left coset ráañ
are highlighted.
Describe the position of the left coset elements in relation to
the subgroup elements in the top row. Explain what you see in
terms of the definition of left coset.
 Click on an element of the subgroup in the leftmost column.
What can you say about this left coset of ráañ?
 How many elements will each left coset of ráañ have?
 Click, in turn, on each of the rest of the elements in the leftmost column of
the Cayley table. What do the different colours show? How many
colours are used?
 Repeat the above steps for the subgroup ábñ.
 Which theorem about a finite group G, and a subgroup H
tells you about the relation between the number of left cosets of
H , the order of H, and the order of G. What does it
say? Choose one other group, that the applet provides and choose
another subgroup. Find all the left cosets of the subgroup. Check
that the relation between the number of left cosets of H, the
order of H, and the order of G is what you would expect.
Activity 2
In this activity you will explore right cosets of subgroups of
S(^{[¯]}) and of subgroups of at least one other group.
The first activity used left cosets. There is an
analagous definition of right coset. We will consider this now.
The right coset Hg of a subgroup H of a group G,
generated by g is defined as Hg = {hg:h Î H}). Can you
see from the Cayley table for
S(^{[¯]}) what the elements of
áañr = {hr:h Î áañ} will be?
 In the applet click on the group
S(^{[¯]}) again. Click
on the subgroup áañ and then click on the element
r in the top row of the Cayley table. Observe how the elements
of the right coset áañr are highlighted.
Describe the position of the right coset elements in relation to
the subgroup elements in the leftmost column. Explain what you see
in terms of the definition of right coset.
 Click on an element of the subgroup áañ in the top row.
What can you say about this right coset?
 How many elements will each coset have?
 Click, in turn, on each of the rest of the elements in the top row of
the Cayley table. What do the different colours show? How many
colours are used?
 Repeat the above steps for the subgroup ábñ.
 Which theorem about a finite group G, and a subgroup H
tells you about the relation between the number of right cosets of
H, the order of H, and the order of G.
 Use the applet to help you find the following for two
different groups:
 a subgroup, whose right and left cosets are the same;
 a subgroup, with different right and left cosets.
 In the previous question, did you rule out some groups? Why?
 In question 7a, was there a choice of subgroup
for which you felt the left and right cosets had to be the same?
Why? [Think about the number of cosets.]
There are two buttons that you have not yet used in the applet.
These say `Left cosets' and `Right cosets'. Choose the group
S(^{[¯]}) and the subgroup áañ. Click on the
`Left cosets' button. Observe that
all the elements have been coloured according to their coset.
Click on the subgroup ábñ, followed by the
`Left cosets' button. Notice that the elements have been
reordered so that cosets are grouped together. Repeat this for
the 'Right cosets' button.
Activity 3
In this activity you will be reminded about
normal subgroups.
 Select the group
S(^{[¯]}) and the
subgroup árñ. Click on the `Left cosets' button.
Use the colours to help you find two pairs of elements x,y and
v,w such that x and v belong to the same coset, y
and w belong to the same coset but xy and vw belong to
different cosets.
Now reset, choose the same group and subgroup and use the applet
to write down the elements of the left coset xárñ
and the right coset árñx. Are they the same?
What about yárñ and árñy?
 Select the group
S(Tri) and the subgroup árñ. Repeat the exercise above. Can you find a subgroup of
S(Tri) for which you cannot repeat the above exercise?
A normal subgroup H of a group G, is a subgroup
whose left and right cosets are the same. Use the applet to help
you find all the proper normal subgroups of
S(^{[¯]}), S(Tri)
and Q. (Proper means not equal to the identity subgroup or the
whole subgroup.) Can you use Theorem 1 in Frame 12 on p29 of Unit
2 to make your task easier?
Activity 4
In this activity you will investigate
some quotient groups.
Click on "Load quotient groups applet" below to load the applet.
(This applet needs more room as it sometimes shows two group
tables. You may find it sensible to view the full screen in your
browser  for internet explorer, click on View in the top menu bar
and select full screen. Use the scroll bar to move up and down the
applet as necessary.)
The last task of the previous section was
made easier by clicking on the `Left cosets' button. If the
resulting table divided into neat squares of colour, all the same
size the subgroup was normal. Otherwise it was not normal. This is
because the binary operation in the group transfers to the cosets
when the subgroup is normal. That is, in a group G with normal
subgroup N, for any two pairs of elements x,y and u,v
such that x and u belong to the same coset of N, and
y and v belong to the same coset of N then xy and
uv belong to the same coset of N. In this case the cosets
of N can be regarded as elements of a new group. We call this
group the quotient group G/N.
 In the applet, select the group S(^{[¯]}) and the subgroup áañ. In this applet the Cayley table is reorganised to show
the cosets as soon as you select the subgroup. In this case the
subgroup is normal. Look at the shape of the table in terms of the
colours:
 blue  red 
blue  blue  red 
red  red  blue

This will be the pattern of the Cayley table for the Quotient
group and is the same shape as the Cayley table for C_{2}. Click
on C_{2} in the quotient group list so that you can compare the
tables. We can see that S(^{[¯]})/áañ is isomorphic
to C_{2}. The identity element in the quotient group is blue and
corresponds to the rotations, the other element is red and
corresponds to reflections. This demonstrates one of the uses of
quotient groups  they enable you to see how sets of elements
relate to each other.
 Can you find a quotient group of S(^{[¯]}) isomorphic to
K?
 What quotient groups of S(Tri) can you find? What does
Lagrange's Theorem tell you about the order of a quotient group of
a finite group?
 Use the applet to help you find the quotient
groups isomorphic to the following groups: C_{3},C_{4},S_{3}. [The
quotient groups should arise from proper normal subgroups.]
 In the applet select the group C_{4}×C_{2}. This is an
abelian group of order 8. Select the subgroup áa^{2}ñ. To which group is this subgroup isomorphic?
To which group is the quotient group isomorphic?
Select the subgroup ábñ. To which group is this
subgroup isomorphic?
To which group is the quotient group isomorphic?
You should have found that
(C_{4}×C_{2})/áa^{2}ñ is isomorphic to K 

and
(C_{4}×C_{2})/ábñ is isomorphic to C_{4}. 

You should also have observed that
áa^{2}ñ isisomorphic to C_{2} is isomorphic to ábñ. 

This gives us an example of a group G, with two isomorphic
normal subgroups H,N such that G/H is not isomorphic to
G/N.
File translated from T_{E}X by T_{T}H, version 2.00.
On 25 Feb 2002, 11:06.