1Groups and homomorphisms

IA Groups



1.2 Homomorphisms
It is often helpful to study functions between different groups. First, we need to
define what a function is. These definitions should be familiar from IA Numbers
and Sets.
Definition
(Function)
.
Given two sets
X
,
Y
, a function
f
:
X Y
sends each
x X
to a particular
f
(
x
)
Y
.
X
is called the domain and
Y
is the co-domain.
Example.
Identity function: for any set
X
, 1
X
:
X X
with 1
X
(
x
) =
x
is a function.
This is also written as id
X
.
Inclusion map:
ι
:
Z Q
:
ι
(
n
) =
n
. Note that this differs from the
identity function as the domain and codomain are different in the inclusion
map.
f
1
: Z Z: f
1
(x) = x + 1.
f
2
: Z Z: f
2
(x) = 2x.
f
3
: Z Z: f
3
(x) = x
2
.
For g : {0, 1, 2, 3, 4} {0, 1, 2, 3, 4}, we have:
g
1
(x) = x + 1 if x < 4; g
1
(4) = 4.
g
2
(x) = x + 1 if x < 4; g
1
(4) = 0.
Definition
(Composition of functions)
.
The composition of two functions is a
function you get by applying one after another. In particular, if
f
:
X Y
and
G : Y Z, then g f : X Z with g f(x) = g(f (x)).
Example. f
2
f
1
(
x
) = 2
x
+ 2.
f
1
f
2
(
x
) = 2
x
+ 1. Note that function
composition is not commutative.
Definition
(Injective functions)
.
A function
f
is injective if it hits everything
at most once, i.e.
(x, y X) f(x) = f(y) x = y.
Definition
(Surjective functions)
.
A function is surjective if it hits everything
at least once, i.e.
(y Y )(x X) f(x) = y.
Definition
(Bijective functions)
.
A function is bijective if it is both injective
and surjective. i.e. it hits everything exactly once. Note that a function has an
inverse iff it is bijective.
Example. ι
and
f
2
are injective but not subjective.
f
3
and
g
1
are neither. 1
X
,
f
1
and g
2
are bijective.
Lemma. The composition of two bijective functions is bijective
When considering sets, functions are allowed to do all sorts of crazy things,
and can send any element to any element without any restrictions. However, we
are currently studying groups, and groups have additional structure on top of
the set of elements. Hence we are not interested in arbitrary functions. Instead,
we are interested in functions that “respect” the group structure. We call these
homomorphisms.
Definition
(Group homomorphism)
.
Let (
G,
) and (
H, ×
) be groups. A
function f : G H is a group homomorphism iff
(g
1
, g
2
G) f(g
1
) × f(g
2
) = f(g
1
g
2
),
Definition
(Group isomorphism)
.
Isomorphisms are bijective homomorphisms.
Two groups are isomorphic if there exists an isomorphism between them. We
write G
=
H.
We will consider two isomorphic groups to be “the same”. For example, when
we say that there is only one group of order 2, it means that any two groups of
order 2 must be isomorphic.
Example.
f
:
G H
defined by
f
(
g
) =
e
, where
e
is the identity of
H
, is a
homomorphism.
1
G
:
G G
and
f
2
:
Z
2
Z
are isomorphisms.
ι
:
Z Q
and
f
2
:
Z Z
are homomorphisms.
exp : (R, +) (R
+
, ×) with exp(x) = e
x
is an isomorphism.
Take (
Z
4
,
+) and
H
: (
{e
ikπ/2
:
k
= 0
,
1
,
2
,
3
}, ×
). Then
f
:
Z
4
H
by
f(a) = e
a/2
is an isomorphism.
f
:
GL
2
(
R
)
R
with
f
(
A
) =
det
(
A
) is a homomorphism, where
GL
2
(
R
)
is the set of 2 × 2 invertible matrices.
Proposition. Suppose that f : G H is a homomorphism. Then
(i) Homomorphisms send the identity to the identity, i.e.
f(e
G
) = e
H
(ii) Homomorphisms send inverses to inverses, i.e.
f(a
1
) = f(a)
1
(iii) The composite of 2 group homomorphisms is a group homomorphism.
(iv) The inverse of an isomorphism is an isomorphism.
Proof.
(i)
f(e
G
) = f(e
2
G
) = f(e
G
)
2
f(e
G
)
1
f(e
G
) = f(e
G
)
1
f(e
G
)
2
f(e
G
) = e
H
(ii)
e
H
= f(e
G
)
= f(aa
1
)
= f(a)f(a
1
)
Since inverses are unique, f(a
1
) = f(a)
1
.
(iii)
Let
f
:
G
1
G
2
and
g
:
G
2
G
3
. Then
g
(
f
(
ab
)) =
g
(
f
(
a
)
f
(
b
)) =
g(f(a))g(f (b)).
(iv) Let f : G H be an isomorphism. Then
f
1
(ab) = f
1
n
f
f
1
(a)
f
f
1
(b)
o
= f
1
n
f
f
1
(a)f
1
(b)
o
= f
1
(a)f
1
(b)
So
f
1
is a homomorphism. Since it is bijective,
f
1
is an isomorphism.
Definition
(Image of homomorphism)
.
If
f
:
G H
is a homomorphism, then
the image of f is
im f = f(G) = {f(g) : g G}.
Definition (Kernel of homomorphism). The kernel of f, written as
ker f = f
1
({e
H
}) = {g G : f(g) = e
H
}.
Proposition.
Both the image and the kernel are subgroups of the respective
groups, i.e. im f H and ker f G.
Proof.
Since
e
H
im f
and
e
G
ker f
,
im f
and
ker f
are non-empty. Moreover,
suppose
b
1
, b
2
im f
. Now
a
1
, a
2
G
such that
f
(
a
i
) =
b
i
. Then
b
1
b
1
2
=
f(a
1
)f(a
1
2
) = f(a
1
a
1
2
) im f.
Then consider
b
1
, b
2
ker f
. We have
f
(
b
1
b
1
2
) =
f
(
b
1
)
f
(
b
2
)
1
=
e
2
=
e
. So
b
1
b
1
2
ker f.
Proposition.
Given any homomorphism
f
:
G H
and any
a G
, for all
k ker f, aka
1
ker f.
This proposition seems rather pointless. However, it is not. All subgroups
that satisfy this property are known as normal subgroups, and normal subgroups
have very important properties. We will postpone the discussion of normal
subgroups to later lectures.
Proof. f(aka
1
) = f(a)f(k)f (a)
1
= f(a)ef(a)
1
= e. So aka
1
ker f.
Example. Images and kernels for previously defined functions:
(i) For the function that sends everything to e, im f = {e} and ker f = G.
(ii) For the identity function, im 1
G
= G and ker 1
G
= {e}.
(iii) For the inclusion map ι : Z Q, we have im ι = Z and ker ι = {0}
(iv) For f
2
: Z Z and f
2
(x) = 2x, we have im f
2
= 2Z and ker f
2
= {0}.
(v)
For
det
:
GL
2
(
R
)
R
, we have
im det
=
R
and
ker det
=
{A
:
det A
=
1} = SL
2
(R)
Proposition. For all homomorphisms f : G H, f is
(i) surjective iff im f = H
(ii) injective iff ker f = {e}
Proof.
(i) By definition.
(ii)
We know that
f
(
e
) =
e
. So if
f
is injective, then by definition
ker f
=
{e}
. If
ker f
=
{e}
, then given
a, b
such that
f
(
a
) =
f
(
b
),
f
(
ab
1
) =
f(a)f(b)
1
= e. Thus ab
1
ker f = {e}. Then ab
1
= e and a = b.
So far, the definitions of images and kernels seem to be just convenient
terminology to refer to things. However, we will later prove an important
theorem, the first isomorphism theorem, that relates these two objects and
provides deep insights (hopefully).
Before we get to that, we will first study some interesting classes of groups
and develop some necessary theory.