From: Beachy/Blair,

Covering: Sections 3.1 through 3.6

(i)

(ii)

(iii)

(iv)

**
3.1.6. Proposition.
**
(Cancellation Property for Groups)
Let * G * be a group, and let * a,b,c *
be elements of * G*.

(a)
If * ab = ac *, then * b = c *.

(b)
If * ac = bc *, then * a = b *.

**
3.1.8. Definition.
**
A group * G * is said to be
** abelian **
if * a b = b a * for all * a,b * in * G*.

**
3.1.9. Definition.
**
A group * G * is said to be a
** finite ** group
if the set * G * has a finite number of elements.
In this case, the number of elements is called the
** order **
of * G*, denoted by
| * G * |.

**
3.2.7. Definition.
**
Let *a* be an element of the group * G*.
If there exists a positive integer * n *
such that *a*^{n} = *e*,
then * a * is said to have
** finite order**,
and the smallest such positive integer is called the
** order**
of * a *, denoted by
* o(a) *.

If there does not exist a positive integer * n * such
that *a*^{n} = *e*, then * a * is said to have
** infinite order**.

**
3.2.1. Definition.
**
Let * G * be a group,
and let * H * be a subset of * G*.
Then * H * is called a
** subgroup **
of * G * if * H * is itself a group,
under the operation induced by * G *.

**
3.2.2. Proposition.
**
Let * G * be a group with identity element * e *, and
let * H * be a subset of * G*.
Then * H * is a subgroup of * G * if and only if
the following conditions hold:

(i)
* ab * is in * H * for all * a,b * in * H*;

(ii)
* e * is in * H*;

(iii)
for all * a * in * H*,
the inverse of * a * is also in * H*.

**
3.2.10. Theorem.
(Lagrange)
**
If * H * is a subgroup of the finite group * G*, then
the order of * H * is a divisor of the order of * G*.

**
3.2.11. Corollary.
**
Let * G * be a finite group of order * n *.

(a)
For any * a * in * G*,
* o(a) * is a divisor of * n *.

(b)
For any * a * in * G*, *a*^{n} = *e*.

**
3.2.12. Corollary.
**
Any group of prime order is cyclic.

**
3.4.1. Definition.
**
Let * G * and * G' * be groups,
and let * f : G * -> * G' * be a function.
Then * f * is said to be a
** group isomorphism **
if * f * is one-to-one and onto and

* f (ab) = f (a) f (b) *

for all * a,b * in * G*.
In this case, * G* is said to be
** isomorphic ** to * G'*.

**
3.4.3. Proposition.
**
Let * f : G * -> * G' * be an isomorphism of groups.

(a)
If * a * has order * n * in * G*,
then * f (a) * has order * n * in * G'*.

(b)
If * G * is abelian, then so is * G'*.

(c)
If * G * is cyclic, then so is * G'*.

is called the

The group

**
3.2.6 Proposition.
**
Let * G * be a group,
and let * a * be in * G*.

(a)
The set * < a > * is a subgroup of * G*.

(b)
If * K * is any subgroup of * G *
such that * a * in * K*,
then * < a > * is a subset of * K*.

**
3.2.8. Proposition.
**
Let * a * be an element of the group * G*.

(a)
If * a * has infinite order,
and *a*^{k} = *a*^{m}
for integers * k , m *,
them * k = m *.

(b)
If * a * has finite order and * k * is any integer,
then *a*^{k} = *e*
if and only if * o(a) * | * k *.

(c)
If * a * has finite order * o(a) = n *,
then for all integers * k,m *,
we have *a*^{k} = *a*^{m} if and only if
* k * is congruent to * m * modulo * n *.
Furthermore, |* < a > *| * = o(a) *.

**
Corollaries to Lagrange's Theorem
**
(restated):

(a)
For any * a * in *G*,
* o(a) * is a divisor of | * G * |.

(b)
For any * a * in * G*, *a*^{n} = *e*,
for * n = * | * G * |.

(c)
Any group of prime order is cyclic.

**
3.5.1. Theorem.
**
Every subgroup of a cyclic group is cyclic.

**
3.5.2 Theorem.
**
Let * G * be a cyclic group.

(a)
If * G * is infinite,
then * G * is isomorphic to the group
* Z * of integers.

(b) If |

**
3.5.3. Proposition.
**
Let * G = < a > * be a cyclic group with order * n*.

(a)
If * m * is any integer,
then < *a*^{m} > = < *a*^{d} > ,
where * d = * gcd (*m,n*),
and *a*^{m} has order * n / d *.

(b)
The element *a*^{k} generates * G *
if and only if gcd (*k,n*) = 1.

(c)
The subgroups of * G * are in one-to-one correspondence with
the positive divisors of * n *.

(d)
If * m * and * k * are divisors of * n * ,
then < *a*^{m} > is a subset of
< *a*^{k} >
if and only if * k * | * m * .

The set of all permutations of the set {

**
3.1.5. Proposition.
**
If * S * is any nonempty set, then Sym(*S*) is a group
under the operation of composition of functions.

**
2.3.5. Theorem.
**
Every permutation in * S*_{n} can be written as a product
of disjoint cycles.
The cycles that appear in the product are unique.

**
2.3.8 Proposition.
**
If a permutation in * S*_{n}
is written as a product of disjoint cycles,
then its order is the least common multiple of the
lengths of its cycles.

**
3.6.1. Definition.
**
Any subgroup of the symmetric group Sym(*S*)
on a set * S * is called a
** permutation group**.

**
3.6.2. Theorem.
(Cayley)
**
Every group is isomorphic to a permutation group.

**
3.6.3. Definition.
**
Let * n > 2 * be an integer.
The group of rigid motions of a regular *n*-gon is
called the *n*th
** dihedral group**,
denoted by * D*_{n}.

We can describe *D*_{n} as
{ * a ^{k}, a^{k} b * | 0

**
2.3.11. Theorem.
**
If a permutation is written as a product of transpositions in two ways,
then the number of transpositions is either even in both cases
or odd in both cases.

**
2.3.12. Definition.
**
A permutation is called
** even **
if it can be written as a product of an even number of transpositions, and
** odd **
if it can be written as a product of an odd number of transpositions.

**
3.6.4. Proposition.
**
The set of all even permutations of * S*_{n}
is a subgroup of * S*_{n}.

**
3.6.5. Definition.
**
The set of all even permutations of * S*_{n} is called the
** alternating group **
on * n * elements, and will be denoted by
* A*_{n}.

**
3.1.10. Definition.
**
The set of all invertible * n x n * matrices with
entries in * R * is called the

**
3.1.11. Proposition.
**
The set * GL*_{n} (* R*)
forms a group under matrix multiplication.

**
3.3.3. Definition.
**
Let * G * and * G' * be groups.
The set of all ordered pairs ( *a , a' * ) such that
* a * is in * G * and
* a' * is in * G' * is called the
** direct product **
of * G * and * G' *, denoted by

* G * x * G' * .

**
3.3.4. Proposition.
**
Let * G * and * G' * be groups.

(a)
The direct product * G* x * G' *
of * G * and * G' *
is a group under the multiplication

( *a , a' * )
( *b , b' * ) =
( *a b , a' b' * ) .

(b)
If the elements * a * in * G * and
* a' * in * G' * have orders
* n * and * m *, respectively, then in

* G * x * G' * the
element ( *a , a' * ) has order lcm [*n,m*].

**
3.4.5. Proposition.
**
If * m,n * are positive integers such that gcd (*m,n*) = 1 ,
then the direct product

(**Z**_{m}) x (**Z**_{n})
is isomorphic to **Z**_{mn}.

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