From: Beachy/Blair,

Covering: Sections 3.7 and 3.8

is called the

The number of left cosets of

**
Proposition 3.8.1.
**
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 * is a subset of * aH*;

(3) * b * is in * aH*;

(4) * a*^{-1}*b * is in * 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, b * be elements of * G *.
Then the following conditions are equivalent:

(1) * Ha = Hb*;

(2) * Ha * is a subset of * Hb*;

(3) * a * is in * Hb*;

(4) * ab*^{-1} is in * H*;

(5) * ba*^{-1} is in * H*;

(6) * b * is in * Ha *;

(7) * Hb * is a subset of * 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.

**
Definition 3.7.5.
**
A subgroup * H* of the group * G* is called a
** normal **
subgroup if * ghg*^{-1} is in * H *
for all * h * in * H * and all * g * in *G*.

**
Proposition 3.8.7.
**
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 * in * G*;

(3) for all * a,b * in * G*,
the coset * abH * is the set theoretic product * (aH)(bH)*;

(4) for all * a,b * in * G*,
the product * ab*^{-1} is in * H * if and only
if * a*^{-1}*b * is in * H*.

**
Example 3.8.7.
**
Any subgroup of index * 2* is normal.

If

**
Theorem 3.8.4.
**
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 * in * G*.

**
Definition 3.8.5.
**
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 * is in * G*, then the order of * aN *
is the smallest positive integer * n *
such that * a ^{n} * is in

**
Example 3.7.1.
**
(Exponential functions for groups)
Let * G * be any group,
and let * a * be any element of * G*.
Define * f : Z * ->

If

**
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.

**
Proposition 3.7.2.
**
If * f : G * -> * G' * is a group homomorphism, then

(a) * f * maps identity to identity, so
* f(e) = e*;

(b) * f * preserves inverses, so
* (f(a))*^{-1}* = f(a*^{-1})
for all * a * in * G*;

(c) * f * preserves powers, so
for any integer * n * and any * a * in * G*,
* f(a ^{n}) = (f(a))^{n}*;

(d) if

**
Example 3.7.4.
**
(Homomorphisms defined on cyclic groups)
Let * C* be a cyclic group, denoted multiplicatively,
with generator * a*.
If * f : C * -> * G* is any group homomorphism,
and * f(a) = g*, then
the formula * f(a ^{m}) = g^{m}* must hold.
Since every element of

If

If |

**
Example 3.7.5.
**
(Homomorphisms from **Z**_{n}
to **Z**_{k})
Any homomorphism
* f : Z*

**
Definition 3.7.3
**
Let * f : G * -> * G' * be a group homomorphism.
Then

{ * x in G * | * f(x) = e * }

is called the
** kernel **
of * f *, and is denoted by ker (*f*).

**
Proposition 3.7.4
**
Let * f : G * -> * G * be a group homomorphism,
with * K = * ker (*f*).

(a) * K * is a normal subgroup of * G*.

(b) The homomorphism * f * is one-to-one
if and only if * K = * {*e*}.

**
Proposition 3.7.6
**
Let * f : G * -> * G' * be a group homomorphism.

(a) If * H * is a subgroup of * G*,
then * f(H) * is a subgroup of * G' *.

If * f * is onto and * H * is normal in * G*,
then * f(H) * is normal in * G'*.

(b) If * H' * is a subgroup of * G'*,
then the inverse image of * H' *
is a subgroup of * G*.

If * H' * is a normal in * G'*, then
the inverse image of * H' * is normal in * G*.

**
Proposition 3.8.6.
**
Let * N * be a normal subgroup of * G*.

(a) The natural projection mapping * p : G * -> * G / N *
defined by * p(x) = xN *, for all * x * in * G*,
is a homomorphism, and ker (*p*) = * 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
**Z**_{n} / m **Z**_{n}
is isomorphic to **Z**_{m}.

**
Theorem 3.8.8.
[Fundamental Homomorphism Theorem]
**
Let * G *, * G' * be groups.
If * f : G * -> * G' * is a group homomorphism
with * K * = ker (*f*),
then * G / K * is isomorphic to the image * f(G)*.

**
Definition 3.8.9.
**
The group * G* is called a
** simple **
group if it has no proper nontrivial normal subgroups.

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