Cosets and normal subgroups
3.8.2. Definition Let H be a subgroup of the group G, and let a
 G. The set
aH = { x
G | x = ah
for some h H
}
is called the left coset of H in G determined by a. Similarly, the
right coset of H in G determined by a is the set
Ha = { x
G | x = ha
for some h H
}.
The number of left cosets of H in G is called the index of H in G, and is
denoted by [G:H].
3.8.1. Proposition Let H be a subgroup of the group G, and let a,b be
elements of G. Then the following conditions are equivalent:
- (1) bH = aH;
- (2) bH
 aH;
- (3) b
aH;
- (4) a-1b
H.
A result similar to Proposition 3.8.1 holds for right cosets. Let H be a
subgroup of the group G, and let a,bG.
Then the following conditions are equivalent:
- (1) Ha = Hb; (2) Ha
Hb;
(3) a Hb;
(4) ab-1
H;
- (5) ba-1
H; (6) b
Ha; (7)
Hb
Ha.
The index of H in G could also be defined as the number of right cosets of H in
G, since there is a one-to-one correspondence between left cosets and right
cosets.
3.7.5. Definition A subgroup H of the group G is called a normal
subgroup if
ghg-1
H
for all h H
and g G.
3.8.7. Proposition Let H be a subgroup of the group G. The following
conditions are equivalent:
- (1) H is a normal subgroup of G;
- (2) aH = Ha for all a
G;
- (3) for all a,b
G, abH is
the set theoretic product (aH)(bH);
- (4) for all a,b
G, ab-1
H if and
only if a-1b
H.
Example 3.8.7. Any subgroup of index 2 is normal.
Factor groups
3.8.3. Proposition Let N be a normal subgroup of G, and let a,b,c,d
G.
If aN = cN and bN = dN, then abN = cdN.
3.8.4. Theorem If N is a normal subgroup of G, then the set of left
cosets of N forms a group under the coset multiplication given by
aNbN = abN
for all a,b
G.
3.8.5. Definition If N is a normal subgroup of G, then the group of
left cosets of N in G is called the factor group of G determined by N. It
will be denoted by G/N.
Example 3.8.5. Let N be a normal subgroup of G. If a
G, then the
order of aN in G/N is the smallest positive integer n such that an
N.
Group homomorphisms
3.7.1. Definition Let G1 and G2 be groups, and let
 : G1
-> G2 be a function. Then
is said to
be a group homomorphism if
(ab) =
(a)
(b) for all
a,b G1.
Example 3.7.1. (Exponential functions for groups) Let G be any group, and
let a be any element of G. Define
: Z
-> G by (n)
= an, for all n
Z.
This is a group homomorphism from Z to G.
If G is abelian, with its operation denoted additively, then we define
: Z
-> G by (n)
= na.
Example 3.7.2. (Linear transformations) Let V and W be vector spaces.
Since any vector space is an abelian group under vector addition, any linear
transformation between vector spaces is a group homomorphism.
3.7.2. Proposition If
: G1
-> G2 is a group homomorphism, then
- (a)
(e) =
e;
- (b) ((a))-1
= (a-1)
for all a G
1;
- (c) for any integer n and any a
G1,
we have (an)
= ((a))n;
- (d) if a
G1
and a has order n, then the order of
(a) in
G2 is a divisor of n.
Example 3.7.4. (Homomorphisms defined on cyclic groups) Let C be a cyclic
group, denoted multiplicatively, with generator a. If
: C -> G
is any group homomorphism, and
(a) = g,
then the formula
(am)
= gm must hold. Since every element of C is of the form am
for some integer m, this means that
is
completely determined by its value on a.
If C is infinite, then for an element g of any group G, the formula
(am)
= gm defines a homomorphism.
If |C|=n and g is any element of G whose order is a divisor of n, then the
formula (am)
= gm defines a homomorphism.
Example 3.7.5. (Homomorphisms from Zn to Zk)
Any homomorphism
: Zn
-> Zk is completely determined by
([1]n),
and this must be an element [m]k of Zk whose order
is a divisor of n. Then the formula
([x]n)
= [mx]k, for all [x]n
Zn,
defines a homomorphism. Furthermore, every homomorphism from Zn
into Zk must be of this form. The image
(Zn)
is the cyclic subgroup generated by [m]k.
3.7.3 Definition Let
: G1
-> G2 be a group homomorphism. Then
{ x
G1
| (x) = e }
is called the kernel of
, and is
denoted by ker().
3.7.4 Proposition Let
: G1
-> G2 be a group homomorphism, with K = ker().
- (a) K is a normal subgroup of G.
- (b) The homomorphism
is
one-to-one if and only if K = {e}.
3.7.6 Proposition Let
: G1
-> G2 be a group homomorphism.
- (a) If H1 is a subgroup of G1, then
(H1)
is a subgroup of G2.
If is
onto and H1 is normal in G1, then
(H1)
is normal in G2.
- (b) If H2 is a subgroup of G2, then
-1
(H2) = { x
G1
| (x)
H2
}
is a subgroup of G1.
If H2 is normal in G2, then
-1(H2)
is normal in G1.
3.8.6. Proposition Let N be a normal subgroup of G.
- (a) The natural projection mapping
 : G ->
G/N defined by
(x) = xN,
for all x
G, is a homomorphism, and ker()
= N.
- (b) There is a one-to-one correspondence between subgroups of G/N
and subgroups of G that contain N. Under this correspondence, normal
subgroups correspond to normal subgroups.
Example 3.8.8. If m is a divisor of n, then Zn / mZn
 Zm.
3.8.8. Theorem [Fundamental Homomorphism Theorem] Let G1, G2
be groups.
If : G1
-> G2 is a group homomorphism with K = ker(),
then
G1/K
(G1).
3.8.9. Definition The group G is called a simple group if it has
no proper nontrivial normal subgroups.
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