- Groups, in general
- Cyclic groups
- Permutation groups
- Other examples
- Cosets and normal subgroups
- Factor groups
- Group homomorphisms
- Some group multiplication tables

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**(i)***Closure:*For all a,b G the element a**·**b is a uniquely defined element of G.**(ii)***Associativity:*For all a,b,c G, we havea

**·**(b**·**c) = (a**·**b)**·**c.**(iii)***Identity:*There exists an**identity element**e G such thate

for all a G.**·**a = a and a**·**e = a**(iv)***Inverses:*For each a G there exists an**inverse**element a^{-1}G such thata

**·**a^{-1}= e and a^{-1}**·**a = e.

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

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

**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 H for all a,b H;**(ii)**e H;**(iii)**a^{-1}H for all a H.

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

**(a)**For any a G, o(a) is a divisor of n.**(b)**For any a G, a^{n}= e.

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

**3.4.1. 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 isomorphism**
if

**(i)**is one-to-one and onto and**(ii)**(ab) = (a) (b) for all a,b G_{1}.

**3.4.3. Proposition.**
Let
: G_{1} -> G_{2}
be an isomorphism of groups.

**(a)**If a has order n in G_{1}, then (a) has order n in G_{2}.**(b)**If G_{1}is abelian, then so is G_{2}.**(c)**If G_{1}is cyclic, then so is G_{2}.

<a> = { x G |
x = a^{n}
for some n Z }

The group G is called a

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

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

**(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 havea

Furthermore, |<a>|=o(a).^{k}= a^{m}if and only if k m (mod n).

- (a) For any a G,
o(a) is a divisor of |G|.
- (b) For any a G,
a
^{n}= e, for n = |G|. - (c) Any group of prime order is cyclic.

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

**(a)**If G is infinite, then G**Z**.**(b)**If |G| = n, then G**Z**_{n}.

**(a)**If m**Z**, 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}> <a^{k}> if and only if k | m.

**3.5.7. Lemma.**
Let G be a group, and let a,b G
be elements such that ab = ba.
If the orders of a and b are relatively prime, then o(ab) = o(a)o(b).

**3.5.8. Proposition.**
Let G be a finite abelian group.

**(a)**The exponent of G is equal to the order of any element of G of maximal order.**(b)**The group G is cyclic if and only if its exponent is equal to its order.

The set of all permutations of the set {1,2,...,n} is denoted by S

**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** or
**group of permutations**.

**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 the nth dihedral group as

D_{n}=
{ a^{k},
a^{k}b |
0 k < n },

**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 × n
matrices with entries in **R** is called the
**general linear group**
of degree n over the real numbers, and is denoted by
GL_{n}(**R**).

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

**3.3.3. Definition.**
Let G_{1}
and G_{2}
be groups. The set of all ordered pairs
(x_{1},x_{2})
such that
x_{1} G_{1}
and
x_{2} G_{2}
is called the
**direct product**
of G_{1}
and G_{2}, denoted by
G_{1} × G_{2}.

**3.3.4. Proposition.**
Let G_{1}
and G_{2}
be groups.

**(a)**The direct product G_{1}× G_{2}is a group under the multiplication defined for all

(a_{1},a_{2}), (b_{1},b_{2}) G_{1}× G_{2}by(a

_{1},a_{2}) (b_{1},b_{2}) = (a_{1}b_{1},a_{2}b_{2}).**(b)**If the elements a_{1}G_{1}and a_{2}G_{2}have orders n and m, respectively, then in

G_{1}× G_{2}the element (a_{1},a_{2}) has order lcm[n,m].

**(i)**the set of all elements of F is an abelian group under +;

**(ii)**the set of all nonzero elements of F is an abelian group under**·**;**(iii)**a**·**(b+c) = a**·**b + a**·**c for all a,b,c in F.

**3.3.7. Proposition.**
Let F be a field.
Then GL_{n}(F)
is a group under matrix multiplication.

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

**Z**_{m}
×
**Z**_{n}
**Z**_{mn}.

Consider the following set of invertible 2 × 2 matrices with entries in the field of complex numbers.

± , ± , ± , ± .

If we let
**1** = ,
**i** = ,
**j** = ,
**k** =

**i**^{2} =
**j**^{2} =
**k**^{2} = -**1**;

**i****j** = **k**,
**j****k** = **i**,
**k****i** = **j**;

**j****i** = -**k**,
**k****j** = -**i**,
**i****k** = -**j**.