Quotient Groups
The elements of a group
can be partitioned, using a subgroup
of
,
into cells called **cosets**. One of the nicest features of
ESG will allow you to investigate these cosets, which
sometimes form a new group called a **factor** or
**quotient group** of .
Before the Lab
You will prove the following theorem for homework:
Theorem 6.1:
Let
be a group and
a subgroup of
.
Define
by saying
if and only if
.
Then
is an equivalence relation.
We would obtain a similar result if we defined
by saying
if and only if .
In the the theorem, we have
,
for some
In fact, as we let
run through all the elements of
,
we obtain all elements of
that are related to
under the equivalence relation
.
The cell of the partition created by
that contains
is denoted by
and is called the **right coset **of
in
that
contains** **
.
In the second case, we create
,
the **left coset** of
in
that contains** **
.
Why do both
and
contain ?
For the two examples below, we'll consider the subgroups
and
of
.
We'll compute the right and left cosets for each subgroup. Confirm the
computations as you read through the example.
Example 1:

There are two distinct right cosets,
and
,
which form a partition of the elements of .

There are two distinct left cosets,
and
.
These agree with the right cosets:
for every .
Example 2:

There are three distinct right cosets,
,
,
and
There are three distinct left cosets,
,
,
and
.
This time, however, not all of the left and right cosets agree. That is, there
are elements
so that
The distinction between
and
is crucial and at the heart of studying quotient groups. The collection of
cosets forms a new group precisely when the left and right cosets agree. The
problem, it turns out, is in trying to define a binary operation on the cosets.
The **coset operation** is best defined through an example. We
only have the operation defined on
to use. Let's start by using the left cosets of
.
Set
and
.
Choose an element from
and one from
for example,
and
.
Since
,
and
,
we define the coset operation
by
No matter which representative of the cosets
and
we choose, we will obtain the same answer in this case. So we can construct
the Cayley table using the operation
:

Check that this forms a group of order 2. Which coset serves as the identity element?
If we try to do this with the left cosets of
however, we run into problems. Set
,
,
If we compute
using the elements
and
,
we have
But if we do the calculation with
and
,
we obtain
.
Obviously there is a problem here---the operation is not well-defined. In the
lab, we will see how ESG demonstrates this with color.
To summarize, we have the theorem:
Theorem 6.2:
Let
be a group and let
be a subgroup of
.
The cosets of
in
form a group if and only if
for all
When the condition of the theorem holds, we will say that the subgroup
is a **normal **subgroup of
.
The collection of cosets (right or left, of course) is then called the
**quotient **or** factor group**,
mod
,
denoted by
.
In our examples above, the subgroup
is normal but
is not.
It is especially important for you to work out one example carefully on your
own before you use the computer. Do all the computations **by
hand** for question 1.
1.
Consider the subgroups
and
of
Find all the left cosets of the subgroups in .
Find all the right cosets of the subgroups in .
For which subgroups are the left and right cosets equal?
For each subgroup identified in part (c), construct a group table for the
quotient group
What familiar group has the same group table?
Be sure to bring your subgroup lattices to the lab.
In the Lab
Check your answers to question 1 with ESG before
continuing with the problems below. Choose option 3 (Subgroups and
Cosets/Quotients) from the **Group Properties Menu** for
,
and generate each subgroup
.
Answer ``Y'' to the question, ``Would you like to see the left cosets of this
subgroup?'' Look at the coloring of the Cayley table of
grouped by the left cosets of each subgroup. In some cases, you will be asked,
``Would you like to see the quotient table?'' Be sure you understand how the
Cayley table is transformed when you answer ``Y.''
2.
In your own words, explain how you can determine from the table on the
computer screen that the coset operation is well-defined or not well-defined.
For problems 3-10, you may use the computer for your computations. Answer
the following questions for each group
and **all** nontrivial proper subgroups
of .
Record the **distinct** left and right cosets of the subgroup
in .
Is the subgroup
normal in
?
If so, have the computer construct a group table for the new quotient group
.
What familiar group has the same group table? Your answer should be a known
group from the ESG library. Be sure to record on your
subgroup lattices which subgroups are normal.
Note that there are no general formulas which help us figure out which group
actually is. You have to rely on computing the order of
and on knowing something about the various groups of that order.
3.
4.
5.
6.
7.
8.
9.
10.
Further Work
11.
Make at least two conjectures about the kinds of subgroups which always seem
to be normal in a finite group
.
(Hint: think about the special subgroups that we considered in an earlier lab
or the index of the subgroup).
12.
Make at least two conjectures about the factor groups
,
where
is either the commutator subgroup or center of .
13.
Prove Theorem 6.1: Let
be a group and
a subgroup of
.
Define
by
if and only if
.
Then
is an equivalence relation.
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