Chapter 4

Normal subgroups and quotient

groups

4.1 Group Homomorphisms

Many areas of mathematics involve the study of functions between sets that are of interest. Gen-
erally we are not interested in all functions but only those that interact well with particular prop-
erties or themes - for example in calculus we are usually interested in continuous or maybe differ-
entiable functions - these are the ones to which the principles of calculus apply. In linear algebra,
we don’t study all functions between vector spaces, we study the ones that preserve addition and
multiplcation by scalars, and refer to these as linear transformations. Likewise in group theory, we
are interested in functions between groups that preserve the group operations in the sense of the
following definition.
Definition 4.1.1. Let G and H be groups with operations �G and �H respectively. A function φ : G → H
is a group homomorphism if for all elements x and y of G
φ(x �G y) = φ(x) �H φ(y).
This is saying that φ : G → H is a group homomorphism if for any pair of elements x and y of
G, combining them in G and then applying φ always gives the same result as separately applying
φ to the two of them and then combining their images using the group operation in H.

E XAMPLES OF G ROUP H OMOMORPHISMS

1. The Determinant
Let Q× denote the group of non-zero rational numbers under multiplication, and as usual
let GL(3, Q) denote the group of invertible 3 × 3 matrices with rational entries, under mul-
tiplication.
The function det : GL(3, Q) → Q× that sends every matrix to its determinant is a group
homomorphism, since det(AB) = det(A) det(B) if A and B are 3 × 3 matrices with rational
entries.
2. Let H denote the group {1, −1} under multiplication (the 1 and −1 here are just the ordinary
numbers 1 and −1, H is a group of order 2). We may define a function φ from the group Z
of integers under addition to H by
�
1 if n is even
φ(n) =
−1 if n is odd
Then φ is a group homomorphism. To see this suppose that m and n are elements of Z.
There are four cases to check.
• If m and n are both even then so is m + n and
φ(m)φ(n) = 1 × 1 = 1 = φ(m + n).

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 • If m is even and n is odd then m + n is odd and

φ(m)φ(n) = 1 × (−1) = −1 = φ(m + n).

• If m is odd and n is even then m + n is odd and

φ(m)φ(n) = (−1) × 1 = −1 = φ(m + n).

• If m and n are both odd then m + n is even and

φ(m)φ(n) = (−1) × (−1) = 1 = φn (m + n).

3. The function τ from Z to Z defined for a ∈ Z by τ(a) = 3a is a homomorphism from the

additive group (Z, +) to itself. To see this, note for a, b ∈ Z that

τ(a + b) = 3(a + b) = 3a + 3b = τ(a) + τ(b).

4. The log function (for any base, e.g. 10) is a group homomorphism from the group of positve
real numbers R>0 under multiplication, to the group of all real numbers under addition.
This is saying that for a pair of positive real numbers x and y

log(xy) = log(x) + log(y).

To see this, write a and b respectively for log x and log y. Then

x = 10a , y = 10b =⇒ xy = 10a 10b = 10a+b =⇒ log(xy) = a + b = log x + log y.

Exercise: Show that the function f from Z to Z defined for a ∈ Z by f(a) = a + 1 is not a group
homomorphism (from (Z, +) to itself).

If φ : G → H is a group homomorphism, then there is a subgroup of G and a subgroup of H

naturally associated with φ. These are defined below.
Definition 4.1.2. Suppose that φ : G → H is a homomorphism of groups. Then
1. The kernel of φ is the subset of G consisting of all those elements whose image under φ is idH .

ker φ = {g ∈ G : φ(g) = idH }.

2. The image of φ is the subset of H consisting of all those elements that are the images under φ of
elements of G.
Imφ = {h ∈ H : h = φ(g) for some g ∈ G}.

It is fairly routine to prove that the kernel and image of φ are not only subsets but subgroups
of G and H respectively. This is the content of the next two lemmas.
Lemma 4.1.3. Suppose that φ : G → H is a homomorphism of groups. Then ker φ is a subgroup of G.
Proof. First we show that idG ∈ ker φ. Let g ∈ G and let h = φ(g) in H. Then

h = φ(g) = φ(idG �G g) = φ(idG ) �H φ(g) = φ(idG ) �H h.

Thus φ(idG ) is an element of H that satifies

h = φ(idG ) �H h

for some element h of H. Multiplying both sides of the above equation on the right by h−1 , it
follows that φ(idG ) = idH and hence that idG ∈ ker φ.
Now suppose that g1 , g2 ∈ ker φ. Then

φ(g1 �G g2 ) = φ(g1 ) �H φ(g2 ) = idH �H idH = idH .

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Hence g1 �G g2 ∈ ker φ and ker φ is closed uder the operation of G.
Finally let g ∈ ker φ. We need to show that g−1 ∈ ker φ aswell. We know that φ(g) = idH and
(from above) that φ(idG ) = idH . Now

idH = φ(idG )
= φ(g �G g−1 )
= φ(g) �H φ(g−1 )
= idH �H φ(g−1 )
= φ(g−1 ).

Thus g−1 ∈ ker φ, as required.

Remark: Note that the last part of the above proof shows that φ(g) and φ(g−1 ) are inverses of
each other in H, for any element g of G.
Lemma 4.1.4. Suppose that φ : G → H is a homomorphism of groups. Then Imφ is a subgroup of H.
Proof. From the proof of Lemma 4.1.3 above we know that φ(idG ) = idH , so idH ∈ Imφ.
Suppose that h1 , h2 ∈ Imφ. Then h1 = φ(g1 ) and h2 = φ(g2 ) for some elements g1 and g2 of G.
Then
φ(g1 �G g2 ) = φ(g1 ) �H φ(g2 ) = h1 �H h2 ,
so h1 �H h2 ∈ Imφ and Imφ is closed under �H .
Finally suppose h ∈ Imφ. We need to show that h−1 ∈ Imφ also. We know that h = φ(g) for
some g ∈ G, and it then follows from the Remark above that h−1 = φ(g−1 ).

Examples
1. The Determinant
The kernel of the function det : GL(3, Q) → Q× that sends every matrix to its determinant is
the subgroup consisting of all those matrices of determinant 1 in GL(3, Q). This is denoted
SL(3, Q) and called the special linear group of 3 × 3 matrices over Q.
The image of det is the full group Q× of non-zero rational numbers, since every non-zero
rational number arises as the determinant of some 3 × 3 matrix with rational entries.
2. The “parity function” for integers
Let φ : Z → {1, −1} be defined by by
�
1 if n is even
φ(n) =
−1 if n is odd

Then the kernel of φ is 2Z, the group of even integers in Z. The image of φ is {1, −1}.
3. The homomorphism τ from (Z, +) to (Z, +) defined for a ∈ Z by τ(a) = 3a has trivial kernel
{0}, and its image is the subgroup of (Z, +) consisting of all multiples of 3.
Suppose that group φ : G → H is a surjective group homomorphism. This means that the
image of φ is all of H. If in addition the kernel of φ consists only of the identity element, it means
that φ is injective also. In this situtation φ is called an isomorphism from G to H

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