22.
Quotient groups I
22.1.
Definition of quotient groups.
Let
G
be a group and
H
a subgroup
of
G
.
Denote by
G/H
the set of distinct (left) cosets with respect to
H
.
In other words, we list all the cosets of the form
gH
(with
g
∈
G
) without
repetitions and consider each coset as a SINGLE element of the newly formed
set
G/H
. The set
G/H
(pronounced as
G
mod
H
) is called the quotient set
.
Next we would like to define a binary operation
*
on
G/H
such that (
G/H,
*
)
is a group. It is natural to try to define the operation
*
by the formula
gH
*
kH
=
gkH
for all
g, k
∈
G.
(Q)
Before checking group axioms, we need to find out whether
*
is at least well
defined. Our first result shows that
*
is well defined whenever
H
is a normal
subgroup.
Theorem 22.1.
Let
G
be a group and
H
a normal subgroup of
G
. Then
the operation
*
given by (Q) is well defined.
Proof.
We need to show that if
g
1
, g
2
, k
1
, k
2
∈
G
are such that
g
1
H
=
g
2
H
and
k
1
H
=
k
2
H
, then
g
1
k
1
H
=
g
2
k
2
H
.
Recall that Theorem 19.2 (formulated slightly differently) asserts that given
x, y
∈
G
we have
xH
=
yH
⇐⇒
x

1
y
∈
H
. Thus, we need to show the
following implication
if
g

1
1
g
2
∈
H
and
k

1
1
k
2
∈
H,
then (
g
1
k
1
)

1
g
2
k
2
∈
H
(!)
So, assume that
g

1
1
g
2
∈
H
and
k

1
1
k
2
∈
h
. Then there exist
h, h
∈
H
such
that
g

1
1
g
2
=
h
and
k

1
1
k
2
=
h
, and thus
k
2
=
k
1
h
. Hence
(
g
1
k
1
)

1
g
2
k
2
=
k

1
1
g

1
1
g
2
k
2
=
k

1
1
hk
1
h
= (
k

1
1
hk
1
)
h .
Since
H
is normal,
k

1
1
hk
1
∈
H
by Theorem 20.2, and so (
k

1
1
hk
1
)
h
∈
H
.
Thus, we proved implication (!) and hence also Theorem 22.1.
Remark:
1.
The converse of Theorem 22.1 is also true, that is, if the
operation
*
on
G/H
is well defined, then
H
must be normal. This fact is
left as an exercise, but we will not use it in the sequel.
2.
The book uses a different approach to defining the group operation in
quotient groups (eventually, of course, the definition is the same, but initial
justification is different). We could define the operation
*
by setting
gH
*
kH
to be the product of
gH
and
kH
as subsets of
G
(this operation was
1
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introduced in Lecture 19 and will be referred below as subset product
). With
this definition, it is clear that the product is well defined, but it is not clear
whether
G/H
is closed under it, that is, whether the subset product of two
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 Winter '11
 PhanThuongCang
 Group Theory, Normal subgroup, Coset, G/H, Quotient Groups

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