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3.1. Equivalence Classes
Henceforth:
N = {1, 2, 3, . . .}
3.1. Equivalence Classes
Definition
A relation ∼ on a set X is an equivalence relation if
3.1. Equivalence Classes
Definition
A relation ∼ on a set X is an equivalence relation if
1
x ∼ x ∀x ∈ X (reflexive)
3.1. Equivalence Classes
Definition
A relation ∼ on a set X is an equivalence relation if
1
x ∼ x ∀x ∈ X (reflexive)
2
x ∼ y =⇒ y ∼ x (symmetric)
3.1. Equivalence Classes
Definition
A relation ∼ on a set X is an equivalence relation if
1
x ∼ x ∀x ∈ X (reflexive)
2
x ∼ y =⇒ y ∼ x (symmetric)
3
x ∼ y and y ∼ z =⇒ x ∼ z (transitive)
3.1. Equivalence Classes
Definition
A relation ∼ on a set X is an equivalence relation if
1
x ∼ x ∀x ∈ X (reflexive)
2
x ∼ y =⇒ y ∼ x (symmetric)
3
x ∼ y and y ∼ z =⇒ x ∼ z (transitive)
Example
The relation = is an equivalence relation on Q.
E.g. 3/9 = 1/3 = (−24)/(−72).
3.1. Equivalence Classes
Exercise
Determine which of the following are equivalence relations on the
given set.
3.1. Equivalence Classes
Exercise
Determine which of the following are equivalence relations on the
given set.
1
The set of all things in a store; ∼ means “the same price as.”
3.1. Equivalence Classes
Exercise
Determine which of the following are equivalence relations on the
given set.
1
The set of all things in a store; ∼ means “the same price as.”
2
The set of all people; ∼ means “is a full sibling of.”
3.1. Equivalence Classes
Exercise
Determine which of the following are equivalence relations on the
given set.
1
The set of all things in a store; ∼ means “the same price as.”
2
The set of all people; ∼ means “is a full sibling of.”
3
The set of all positive integers; ∼ means “has a common prime
factor with.”
3.1. Equivalence Classes
Exercise
Determine which of the following are equivalence relations on the
given set.
1
The set of all things in a store; ∼ means “the same price as.”
2
The set of all people; ∼ means “is a full sibling of.”
3
The set of all positive integers; ∼ means “has a common prime
factor with.”
4
The set of all positive integers; ∼ means “has the same
remainder when divided by 3.”
3.1. Equivalence Classes
Definition
A partition on a set X is a collection of disjoint subsets of X whose
union is all of X.
3.1. Equivalence Classes
Definition
A partition on a set X is a collection of disjoint subsets of X whose
union is all of X.
Theorem
Let X be a set.
3.1. Equivalence Classes
Definition
A partition on a set X is a collection of disjoint subsets of X whose
union is all of X.
Theorem
Let X be a set.
1
Any equivalence relation ∼ on X forms a partition of X.
3.1. Equivalence Classes
Definition
A partition on a set X is a collection of disjoint subsets of X whose
union is all of X.
Theorem
Let X be a set.
1
Any equivalence relation ∼ on X forms a partition of X.
2
Any partition of X forms an equivalence relation on X.
3.2. Modular Arithmetic
Theorem
Given a ∈ N and b ∈ Z, there exist q, r ∈ Z such that
b = qa + r
where 0 ≤ r < a. Moreover, q and r are unique.
3.2. Modular Arithmetic
Theorem
Given a ∈ N and b ∈ Z, there exist q, r ∈ Z such that
b = qa + r
where 0 ≤ r < a. Moreover, q and r are unique.
Definition
Let m ∈ N. Given a, b ∈ Z, we say a is congruent to b modulo m and
write a ≡ b (mod m) if
m | (a − b)
3.2. Modular Arithmetic
Theorem
Given a ∈ N and b ∈ Z, there exist q, r ∈ Z such that
b = qa + r
where 0 ≤ r < a. Moreover, q and r are unique.
Definition
Let m ∈ N. Given a, b ∈ Z, we say a is congruent to b modulo m and
write a ≡ b (mod m) if
m | (a − b)
Theorem
Given m ∈ N, congruence modulo m is an equivalence relation on Z.
3.2. Modular Arithmetic
Theorem
Given m ∈ N, every integer is congruent to exactly one of the numbers
0, 1, 2, . . . , (m − 1)
modulo m.
3.2. Modular Arithmetic
Theorem
Given m ∈ N, every integer is congruent to exactly one of the numbers
0, 1, 2, . . . , (m − 1)
modulo m.
Definition
The congruence classes of integers modulo m are
[0], [1], [2], . . . , [m − 1]
The set of congruence classes of integers modulo m
{[0], [1], [2], . . . , [m − 1]}
is denoted Zm .
3.2. Modular Arithmetic
Theorem
If a ≡ b (mod m) and c ≡ d (mod m), then:
3.2. Modular Arithmetic
Theorem
If a ≡ b (mod m) and c ≡ d (mod m), then:
1
a + c ≡ b + d (mod m)
3.2. Modular Arithmetic
Theorem
If a ≡ b (mod m) and c ≡ d (mod m), then:
1
a + c ≡ b + d (mod m)
2
ac ≡ bd (mod m)
3.2. Modular Arithmetic
Theorem
If a ≡ b (mod m) and c ≡ d (mod m), then:
1
a + c ≡ b + d (mod m)
2
ac ≡ bd (mod m)
Corollary
The operations of addition and multiplication of congruence classes
of integers modulo m are well defined:
3.2. Modular Arithmetic
Theorem
If a ≡ b (mod m) and c ≡ d (mod m), then:
1
a + c ≡ b + d (mod m)
2
ac ≡ bd (mod m)
Corollary
The operations of addition and multiplication of congruence classes
of integers modulo m are well defined:
1
[a] + [b] = [a + b]
3.2. Modular Arithmetic
Theorem
If a ≡ b (mod m) and c ≡ d (mod m), then:
1
a + c ≡ b + d (mod m)
2
ac ≡ bd (mod m)
Corollary
The operations of addition and multiplication of congruence classes
of integers modulo m are well defined:
1
[a] + [b] = [a + b]
2
[a][b] = [ab]
3.2. Modular Arithmetic
Theorem
If a ≡ b (mod m), then an ≡ bn (mod m) for any n ∈ N.
3.2. Modular Arithmetic
Theorem
If a ≡ b (mod m), then an ≡ bn (mod m) for any n ∈ N.
Example
Find the remainder when 1282 is divided by 13.
3.2. Modular Arithmetic
Theorem
If a ≡ b (mod m), then an ≡ bn (mod m) for any n ∈ N.
Example
Find the remainder when 1282 is divided by 13.
Exercise
3.2. Modular Arithmetic
Theorem
If a ≡ b (mod m), then an ≡ bn (mod m) for any n ∈ N.
Example
Find the remainder when 1282 is divided by 13.
Exercise
1
Find the remainder when 682 is divided by 13.
3.2. Modular Arithmetic
Theorem
If a ≡ b (mod m), then an ≡ bn (mod m) for any n ∈ N.
Example
Find the remainder when 1282 is divided by 13.
Exercise
1
Find the remainder when 682 is divided by 13.
2
Show that a number is divisible by 3 if and only if the sum of its
digits is a multiple of 3.
3.2. Modular Arithmetic
Theorem
If a ≡ b (mod m), then an ≡ bn (mod m) for any n ∈ N.
Example
Find the remainder when 1282 is divided by 13.
Exercise
1
Find the remainder when 682 is divided by 13.
2
Show that a number is divisible by 3 if and only if the sum of its
digits is a multiple of 3.
3
Show that a number is divisible by 9 if and only if the sum of its
digits is a multiple of 9.
3.3. Group Theory
Definition
A group is a set G together with an operation ◦ that satisfies:
3.3. Group Theory
Definition
A group is a set G together with an operation ◦ that satisfies:
1
a, b ∈ G =⇒ a ◦ b ∈ G (closure)
3.3. Group Theory
Definition
A group is a set G together with an operation ◦ that satisfies:
1
a, b ∈ G =⇒ a ◦ b ∈ G (closure)
2
(a ◦ b) ◦ c = a ◦ (b ◦ c) for every a, b, c ∈ G (associative)
3.3. Group Theory
Definition
A group is a set G together with an operation ◦ that satisfies:
1
a, b ∈ G =⇒ a ◦ b ∈ G (closure)
2
(a ◦ b) ◦ c = a ◦ (b ◦ c) for every a, b, c ∈ G (associative)
3
There is an element e ∈ G such that e ◦ a = a = a ◦ e for every
a ∈ G (identity)
3.3. Group Theory
Definition
A group is a set G together with an operation ◦ that satisfies:
1
a, b ∈ G =⇒ a ◦ b ∈ G (closure)
2
(a ◦ b) ◦ c = a ◦ (b ◦ c) for every a, b, c ∈ G (associative)
3
There is an element e ∈ G such that e ◦ a = a = a ◦ e for every
a ∈ G (identity)
4
For every a ∈ G, there is an element x ∈ G such that
a ◦ x = e = x ◦ a (inverses)
3.3. Group Theory
Theorem
Let G be a group.
3.3. Group Theory
Theorem
Let G be a group.
1
The identity element e is unique.
3.3. Group Theory
Theorem
Let G be a group.
1
The identity element e is unique.
2
a ◦ b = a ◦ c =⇒ b = c
3.3. Group Theory
Theorem
Let G be a group.
1
The identity element e is unique.
2
a ◦ b = a ◦ c =⇒ b = c
3
b ◦ a = c ◦ a =⇒ b = c
3.3. Group Theory
Theorem
Let G be a group.
1
The identity element e is unique.
2
a ◦ b = a ◦ c =⇒ b = c
3
b ◦ a = c ◦ a =⇒ b = c
4
The inverse of each element x is unique.
The inverse of x is denoted x−1 .
3.3. Group Theory
Theorem
Let G be a group.
1
The identity element e is unique.
2
a ◦ b = a ◦ c =⇒ b = c
3
b ◦ a = c ◦ a =⇒ b = c
4
The inverse of each element x is unique.
The inverse of x is denoted x−1 .
5
If a, b ∈ G, there exists a unique x ∈ G such that a ◦ x = b and a
unique y ∈ G such that y ◦ a = b.
3.3. Group Theory
Theorem
Let G be a group.
1
The identity element e is unique.
2
a ◦ b = a ◦ c =⇒ b = c
3
b ◦ a = c ◦ a =⇒ b = c
4
The inverse of each element x is unique.
The inverse of x is denoted x−1 .
5
If a, b ∈ G, there exists a unique x ∈ G such that a ◦ x = b and a
unique y ∈ G such that y ◦ a = b.
6
(a ◦ b)−1 = b−1 ◦ a−1
3.3. Group Theory
Theorem
Let G be a group.
1
The identity element e is unique.
2
a ◦ b = a ◦ c =⇒ b = c
3
b ◦ a = c ◦ a =⇒ b = c
4
The inverse of each element x is unique.
The inverse of x is denoted x−1 .
5
If a, b ∈ G, there exists a unique x ∈ G such that a ◦ x = b and a
unique y ∈ G such that y ◦ a = b.
6
(a ◦ b)−1 = b−1 ◦ a−1
7
(a−1 )−1 = a
3.3. Group Theory
Example
3.3. Group Theory
Example
1
Z = (Z, +)
3.3. Group Theory
Example
1
Z = (Z, +)
2
Q, R, C
3.3. Group Theory
Example
1
Z = (Z, +)
2
Q, R, C
3
Q× , R× , C×
3.3. Group Theory
Example
1
Z = (Z, +)
2
Q, R, C
3
Q× , R× , C×
4
Mm,n (R) = (Mm,n (R), +)
3.3. Group Theory
Example
1
Z = (Z, +)
2
Q, R, C
3
Q× , R× , C×
4
Mm,n (R) = (Mm,n (R), +)
5
GLn (R) = {A ∈ Mn (R) | det A 6= 0}
3.3. Group Theory
Example
1
Z = (Z, +)
2
Q, R, C
3
Q× , R× , C×
4
Mm,n (R) = (Mm,n (R), +)
5
GLn (R) = {A ∈ Mn (R) | det A 6= 0}
6
Zn = (Zn , +)
3.3. Group Theory
Example
1
Z = (Z, +)
2
Q, R, C
3
Q× , R× , C×
4
Mm,n (R) = (Mm,n (R), +)
5
GLn (R) = {A ∈ Mn (R) | det A 6= 0}
6
Zn = (Zn , +)
7
Z×
n
3.3. Group Theory
Example
1
Z = (Z, +)
2
Q, R, C
3
Q× , R× , C×
4
Mm,n (R) = (Mm,n (R), +)
5
GLn (R) = {A ∈ Mn (R) | det A 6= 0}
6
Zn = (Zn , +)
7
Z×
n
8
Sn
3.3. Group Theory
Exercise
Compute the group tables for Z6 and S3 . What similarities do you
notice? What differences?
3.3. Group Theory
Exercise
Compute the group tables for Z6 and S3 . What similarities do you
notice? What differences?
Definition
A group G is said to be abelian if
ab = ba
for every a, b ∈ G.
3.3. Group Theory
Exercise
Compute the group tables for Z6 and S3 . What similarities do you
notice? What differences?
Definition
A group G is said to be abelian if
ab = ba
for every a, b ∈ G.
Remark
We often use the shorthand ab for a ◦ b, and when we do, we say we
are using multiplicative notation. When the operation in the group is
addition – as it is in Z and Zn , for example – we’ll use additive
notation and write a + b for a ◦ b.
Question: Which of the groups above are abelian?
3.3. Group Theory
Definition
The number of elements in a group G is called the order of G and is
denoted |G|.
3.3. Group Theory
Definition
The number of elements in a group G is called the order of G and is
denoted |G|.
Remark
All of the above groups are infinite, except for Zn which has order n,
Sn which has order n!, and Z×
n whose order varies depending on n.
3.3. Group Theory
Definition
A subgroup of a group G is a subset of G that is itself a group under
the operation of G.
3.3. Group Theory
Definition
A subgroup of a group G is a subset of G that is itself a group under
the operation of G.
Exercise
Which of the following are subgroups of Z?
3.3. Group Theory
Definition
A subgroup of a group G is a subset of G that is itself a group under
the operation of G.
Exercise
Which of the following are subgroups of Z?
1
{0}
3.3. Group Theory
Definition
A subgroup of a group G is a subset of G that is itself a group under
the operation of G.
Exercise
Which of the following are subgroups of Z?
1
{0}
2
{1}
3.3. Group Theory
Definition
A subgroup of a group G is a subset of G that is itself a group under
the operation of G.
Exercise
Which of the following are subgroups of Z?
1
{0}
2
{1}
3
{0, 1, 2, 3, 4, 5}
3.3. Group Theory
Definition
A subgroup of a group G is a subset of G that is itself a group under
the operation of G.
Exercise
Which of the following are subgroups of Z?
1
{0}
2
{1}
3
{0, 1, 2, 3, 4, 5}
4
{0, 1, 2, 3, . . . }
3.3. Group Theory
Definition
A subgroup of a group G is a subset of G that is itself a group under
the operation of G.
Exercise
Which of the following are subgroups of Z?
1
{0}
2
{1}
3
{0, 1, 2, 3, 4, 5}
4
{0, 1, 2, 3, . . . }
5
{. . . , −4, −2, 0, 2, 4, . . . }
3.3. Group Theory
Definition
A subgroup of a group G is a subset of G that is itself a group under
the operation of G.
Exercise
Which of the following are subgroups of Z?
1
{0}
2
{1}
3
{0, 1, 2, 3, 4, 5}
4
{0, 1, 2, 3, . . . }
5
{. . . , −4, −2, 0, 2, 4, . . . }
6
{. . . , −3, −1, 1, 3, . . . }
3.3. Group Theory
Theorem
A nonempty subset H of a group G is a subgroup if:
3.3. Group Theory
Theorem
A nonempty subset H of a group G is a subgroup if:
1 It is closed under the operation of G:
3.3. Group Theory
Theorem
A nonempty subset H of a group G is a subgroup if:
1 It is closed under the operation of G:
• a, b ∈ H =⇒ ab ∈ H.
3.3. Group Theory
Theorem
A nonempty subset H of a group G is a subgroup if:
1 It is closed under the operation of G:
• a, b ∈ H =⇒ ab ∈ H.
2
It contains inverses:
3.3. Group Theory
Theorem
A nonempty subset H of a group G is a subgroup if:
1 It is closed under the operation of G:
• a, b ∈ H =⇒ ab ∈ H.
2
It contains inverses:
• a ∈ H =⇒ a−1 ∈ H.
3.3. Group Theory
Theorem
A nonempty subset H of a group G is a subgroup if:
1 It is closed under the operation of G:
• a, b ∈ H =⇒ ab ∈ H.
2
It contains inverses:
• a ∈ H =⇒ a−1 ∈ H.
Exercise
What are the subgroups of Z6 ?
3.3. Group Theory
Theorem
Let G be a group and H a subgroup of G. Define a ∼ b if ab−1 ∈ H.
Then ∼ is an equivalence relation on G.
3.3. Group Theory
Theorem
Let G be a group and H a subgroup of G. Define a ∼ b if ab−1 ∈ H.
Then ∼ is an equivalence relation on G.
Definition
The equivalence classes resulting from the equivalence relation
a ∼ b ⇐⇒ ab−1 ∈ H
are called the right cosets of H in G. The right coset of an element
a ∈ G is denoted Ha and is the set
Ha = {ha | h ∈ H}
3.3. Group Theory
Theorem
Let G be a group and H a subgroup of G. Define a ∼ b if ab−1 ∈ H.
Then ∼ is an equivalence relation on G.
Definition
The equivalence classes resulting from the equivalence relation
a ∼ b ⇐⇒ ab−1 ∈ H
are called the right cosets of H in G. The right coset of an element
a ∈ G is denoted Ha and is the set
Ha = {ha | h ∈ H}
Exercise
Find the right cosets of H = {[0], [3], [6], [9]} in Z12 .
3.3. Group Theory
Lemma
Let H be a subgroup of a group G, and let a ∈ G. The function
f : H → Ha defined by
f : h 7→ ha
is bijective.
3.3. Group Theory
Lemma
Let H be a subgroup of a group G, and let a ∈ G. The function
f : H → Ha defined by
f : h 7→ ha
is bijective.
Lagrange’s Theorem
Theorem
Let G be a finite group and H a subgroup of G. Then
|H| |G|.
3.3. Group Theory
Definition
Given a group G and an element a ∈ G we define
an = |a ◦ a ◦{z· · · ◦ a}
n times
We define a0 = e and a−n = (a−1 )n .
3.3. Group Theory
Definition
Given a group G and an element a ∈ G we define
an = |a ◦ a ◦{z· · · ◦ a}
n times
We define a0 = e and a−n = (a−1 )n .
Theorem
Let G be a group and a ∈ G. Then:
3.3. Group Theory
Definition
Given a group G and an element a ∈ G we define
an = |a ◦ a ◦{z· · · ◦ a}
n times
We define a0 = e and a−n = (a−1 )n .
Theorem
Let G be a group and a ∈ G. Then:
1
a−n = (an )−1
3.3. Group Theory
Definition
Given a group G and an element a ∈ G we define
an = |a ◦ a ◦{z· · · ◦ a}
n times
We define a0 = e and a−n = (a−1 )n .
Theorem
Let G be a group and a ∈ G. Then:
1
a−n = (an )−1
2
am ◦ an = am+n
3.3. Group Theory
Definition
Given a group G and an element a ∈ G we define
an = |a ◦ a ◦{z· · · ◦ a}
n times
We define a0 = e and a−n = (a−1 )n .
Theorem
Let G be a group and a ∈ G. Then:
1
a−n = (an )−1
2
am ◦ an = am+n
3
(am )n = amn
3.3. Group Theory
Theorem
Let G be a group and a ∈ G. Then
hai = {an | n inZ}
is a subgroup of G.
3.3. Group Theory
Theorem
Let G be a group and a ∈ G. Then
hai = {an | n inZ}
is a subgroup of G.
Definition
hai is called the cyclic subgoup of G generated by a. If there exists an
element a ∈ G such that G = hai, then we say that G is a cyclic group
and that a is a generator of G.
3.3. Group Theory
Theorem
Let G be a group and a ∈ G. Then
hai = {an | n inZ}
is a subgroup of G.
Definition
hai is called the cyclic subgoup of G generated by a. If there exists an
element a ∈ G such that G = hai, then we say that G is a cyclic group
and that a is a generator of G.
Example
3.3. Group Theory
Theorem
Let G be a group and a ∈ G. Then
hai = {an | n inZ}
is a subgroup of G.
Definition
hai is called the cyclic subgoup of G generated by a. If there exists an
element a ∈ G such that G = hai, then we say that G is a cyclic group
and that a is a generator of G.
Example
1
Z = h1i = h−1i
3.3. Group Theory
Theorem
Let G be a group and a ∈ G. Then
hai = {an | n inZ}
is a subgroup of G.
Definition
hai is called the cyclic subgoup of G generated by a. If there exists an
element a ∈ G such that G = hai, then we say that G is a cyclic group
and that a is a generator of G.
Example
1
Z = h1i = h−1i
2
Zn = h[1]i
3.3. Group Theory
Theorem
Let G be a group and a ∈ G. Then
hai = {an | n inZ}
is a subgroup of G.
Definition
hai is called the cyclic subgoup of G generated by a. If there exists an
element a ∈ G such that G = hai, then we say that G is a cyclic group
and that a is a generator of G.
Example
1
Z = h1i = h−1i
2
Zn = h[1]i
3
Q, R and C are not cyclic.
3.3. Group Theory
Theorem
Let G be a group and a ∈ G. Then
hai = {an | n inZ}
is a subgroup of G.
Definition
hai is called the cyclic subgoup of G generated by a. If there exists an
element a ∈ G such that G = hai, then we say that G is a cyclic group
and that a is a generator of G.
Example
1
Z = h1i = h−1i
2
Zn = h[1]i
3
Q, R and C are not cyclic.
4
Is S3 cyclic?
3.3. Group Theory
Definition
Let G and K be groups. A function f : G → K is called a
homomorphism if it preserves the group structures of G and K: that is,
if
f (ab) = f (a)f (b)
for all a, b ∈ G.
3.3. Group Theory
Definition
Let G and K be groups. A function f : G → K is called a
homomorphism if it preserves the group structures of G and K: that is,
if
f (ab) = f (a)f (b)
for all a, b ∈ G.
Theorem
Let f : G → K be a homomorphism. Then
f (eG ) = eH
3.3. Group Theory
Definition
Let G and K be groups. A function f : G → K is called a
homomorphism if it preserves the group structures of G and K: that is,
if
f (ab) = f (a)f (b)
for all a, b ∈ G.
Theorem
Let f : G → K be a homomorphism. Then
f (eG ) = eH
Exercise
Let g : G → K be a homomorphism and a ∈ G. Prove that
f (a−1 ) = [f (a)]−1