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

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 a 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 b 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 a. In general the left coset gH of a subgroup H of a group G, generated by g is defined as

gH = {gh:h H}.
For the subgroup a in
S([¯]), this means that the left coset for ra, say is the set
{rh:h a}.
Can you see from the Cayley table what the elements will be?

  1. Click on the subgroup a and then click on the element r in the leftmost column of the Cayley table. Observe how the elements of the left coset ra 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.

  2. Click on an element of the subgroup in the leftmost column. What can you say about this left coset of ra?

  3. How many elements will each left coset of ra have?

  4. 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?

  5. Repeat the above steps for the subgroup b.

  6. 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 H}). Can you see from the Cayley table for
S([¯]) what the elements of ar = {hr:h a} will be?

  1. In the applet click on the group
    S([¯]) again. Click on the subgroup a and then click on the element r in the top row of the Cayley table. Observe how the elements of the right coset ar 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.

  2. Click on an element of the subgroup a in the top row. What can you say about this right coset?

  3. How many elements will each coset have?

  4. 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?

  5. Repeat the above steps for the subgroup b.

  6. 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.

  7. Use the applet to help you find the following for two different groups:

    1. a subgroup, whose right and left cosets are the same;
    2. a subgroup, with different right and left cosets.

  8. In the previous question, did you rule out some groups? Why?

  9. 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 a. Click on the `Left cosets' button. Observe that all the elements have been coloured according to their coset. Click on the subgroup b, followed by the `Left cosets' button. Notice that the elements have been re-ordered 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.

  1. Select the group
    S([¯]) and the subgroup r. 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 xr and the right coset rx. Are they the same? What about yr and ry?

  2. Select the group
    S(Tri) and the subgroup r. 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.

  1. In the applet, select the group S([¯]) and the subgroup a. 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:

    bluered
    bluebluered
    redredblue

    This will be the pattern of the Cayley table for the Quotient group and is the same shape as the Cayley table for C2. Click on C2 in the quotient group list so that you can compare the tables. We can see that S([¯])/a is isomorphic to C2. 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.
  2. Can you find a quotient group of S([¯]) isomorphic to K?
  3. 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?
  4. Use the applet to help you find the quotient groups isomorphic to the following groups: C3,C4,S3. [The quotient groups should arise from proper normal subgroups.]

  5. In the applet select the group C4×C2. This is an abelian group of order 8. Select the subgroup a2. To which group is this subgroup isomorphic?

    To which group is the quotient group isomorphic?

    Select the subgroup b. To which group is this subgroup isomorphic?

    To which group is the quotient group isomorphic?

    You should have found that

    (C4×C2)/a2 is isomorphic to K
    and
    (C4×C2)/b is isomorphic to C4.

    You should also have observed that

    a2 isisomorphic to C2 is isomorphic to b.

    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 TEX by TTH, version 2.00.
On 25 Feb 2002, 11:06.