4Quotient groups
IA Groups
4.3 The Isomorphism Theorem
Now we come to the Really Important Theorem
TM
.
Theorem (The Isomorphism Theorem). Let
f
:
G → H
be a group homomor-
phism with kernel K. Then K ◁ G and G/K
∼
=
im f.
Proof.
We have proved that
K ◁ G
before. We define a group homomorphism
θ : G/K → im f by θ(aK) = f(a).
First check that this is well-defined: If a
1
K = a
2
K, then a
−1
2
a
1
∈ K. So
f(a
2
)
−1
f(a
1
) = f(a
−1
2
a
1
) = e.
So f(a
1
) = f(a
2
) and θ(a
1
K) = θ(a
2
K).
Now we check that it is a group homomorphism:
θ(aKbK) = θ(abK) = f(ab) = f(a)f(b) = θ(aK)θ(bK).
To show that it is injective, suppose
θ
(
aK
) =
θ
(
bK
). Then
f
(
a
) =
f
(
b
). Hence
f(b)
−1
f(a) = e. Hence b
−1
a ∈ K. So aK = bK.
By definition,
θ
is surjective since
im θ
=
im f
. So
θ
gives an isomorphism
G/K
∼
=
im f ≤ H.
If
f
is injective, then the kernel is
{e}
, so
G/K
∼
=
G
and
G
is isomorphic to
a subgroup of
H
. We can think of
f
as an inclusion map. If
f
is surjective, then
im f = H. In this case, G/K
∼
=
H.
Example.
(i)
Take
f
:
GL
n
(
R
)
→ R
∗
with
A 7→ det A
,
ker f
=
SL
N
(
R
).
im f
=
R
∗
as for
all
λ ∈ R
∗
,
det
λ 0 · · · 0
0 1 · · · 0
.
.
.
.
.
.
.
.
.
.
.
.
0 0 0 1
=
λ
. So we know that
GL
n
(
R
)
/SL
n
(
R
)
∼
=
R
∗
.
(ii)
Define
θ
: (
R,
+)
→
(
C∗, ×
) with
r 7→ exp
(2
πir
). This is a group homomor-
phism since
θ
(
r
+
s
) =
exp
(2
πi
(
r
+
s
)) =
exp
(2
πir
)
exp
(2
πis
) =
θ
(
r
)
θ
(
s
).
We know that the kernel is
Z ◁ R
. Clearly the image is the unit circle
(S
1
, ×). So R/Z
∼
=
(S
1
, ×).
(iii) G
= (
Z
∗
p
, ×
) for prime
p
= 2. We have
f
:
G → G
with
a 7→ a
2
. This
is a homomorphism since (
ab
)
2
=
a
2
b
2
(
Z
∗
p
is abelian). The kernel is
{±
1
}
=
{
1
, p −
1
}
. We know that
im f
∼
=
G/ ker f
with order
p−1
2
. These
are known as quadratic residues.
Lemma. Any cyclic group is isomorphic to either
Z
or
Z/
(
nZ
) for some
n ∈ N
.
Proof.
Let
G
=
⟨c⟩
. Define
f
:
Z → G
with
m 7→ c
m
. This is a group
homomorphism since
c
m
1
+m
2
=
c
m
1
c
m
2
.
f
is surjective since
G
is by definition
all c
m
for all m. We know that ker f ◁ Z. We have three possibilities. Either
(i) ker f = {e}, so F is an isomorphism and G
∼
=
Z; or
(ii) ker f = Z, then G
∼
=
Z/Z = {e} = C
1
; or
(iii) ker f
=
nZ
(since these are the only proper subgroups of
Z
), then
G
∼
=
Z/(nZ).
Definition (Simple group). A group is simple if it has no non-trivial proper
normal subgroup, i.e. only {e} and G are normal subgroups.
Example.
C
p
for prime
p
are simple groups since it has no proper subgroups at
all, let alone normal ones. A
5
is simple, which we will prove after Chapter 6.
The finite simple groups are the building blocks of all finite groups. All finite
simple groups have been classified (The Atlas of Finite Groups). If we have
K ◁ G
with
K
=
G
or
{e}
, then we can “quotient out”
G
into
G/K
. If
G/K
is not simple, repeat. Then we can write
G
as an “inverse quotient” of simple
groups.