GROUPS
Excerpted from Beachy/Blair,
Abstract Algebra, 2nd Ed., © 1996
Sections 3.7 and 3.8
 Cosets and normal subgroups
 Factor groups
 Group homomorphisms
 Some group multiplication tables
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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^{1}b 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 onetoone 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^{1}b 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
a^{n} N.
Group homomorphisms
3.7.1. Definition
Let G_{1}
and G_{2}
be groups, and let
: G_{1} > G_{2}
be a function.
Then
is said to be a
group homomorphism if
(ab) =
(a)
(b)
for all a,b G_{1}.
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) = a^{n},
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
: G_{1} > G_{2}
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 G_{1}, we have
(a^{n})
=
((a))^{n};
 (d)
if a G_{1}
and a has order n, then the order of
(a)
in G_{2}
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
(a^{m}) = g^{m}
must hold. Since every element of C is of the form
a^{m}
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
(a^{m}) = g^{m}
defines a homomorphism.
If C=n and g is any element of G whose order is a divisor of n,
then the formula
(a^{m}) = g^{m}
defines a homomorphism.
Example 3.7.5.
(Homomorphisms from
Z_{n}
to Z_{k})
Any homomorphism
: Z_{n} > Z_{k}
is completely determined by
([1]_{n}),
and this must be an element
[m]_{k}
of Z_{k}
whose order is a divisor of n.
Then the formula
([x]_{n}) = [mx]_{k},
for all
[x]_{n}
Z_{n},
defines a homomorphism.
Furthermore, every homomorphism from
Z_{n}
into Z_{k}
must be of this form. The image
(Z_{n})
is the cyclic subgroup generated by
[m]_{k}.
3.7.3 Definition
Let
: G_{1} > G_{2}
be a group homomorphism. Then
{ x
G_{1} 
(x) = e }
is called the kernel of
,
and is denoted by
ker().
3.7.4 Proposition
Let
: G_{1} > G_{2}
be a group homomorphism, with
K = ker().
 (a)
K is a normal subgroup of G.
 (b)
The homomorphism
is onetoone if and only if K = {e}.
3.7.6 Proposition
Let
: G_{1} > G_{2}
be a group homomorphism.
 (a)
If H_{1}
is a subgroup of G_{1}, then
(H_{1})
is a subgroup of G_{2}.
If
is onto and H_{1}
is normal in G_{1}, then
(H_{1})
is normal in G_{2}.
 (b)
If H_{2}
is a subgroup of G_{2}, then
^{1}
(H_{2}) =
{ x
G_{1} 
(x)
H_{2} }
is a subgroup of G_{1}.
If
H_{2}
is normal in G_{2}, then
^{1}(H_{2})
is normal in G_{1}.
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 onetoone 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
Z_{n} / mZ_{n}
Z_{m}.
3.8.8. Theorem [Fundamental Homomorphism Theorem]
Let G_{1},
G_{2} be groups.
If
: G_{1} > G_{2}
is a group homomorphism with
K = ker(), then
G_{1}/K
(G_{1}).
3.8.9. Definition
The group G is called a
simple
group if it has no proper nontrivial normal subgroups.
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