MAS 4301H Honors Modern Algebra
1. N OTATION
Notation 1. Here is a list of some common notation that we will be using throughout the course.
(1) The notation ∈ should be read as either “an element of” or “in”. E.g. x ∈ A can be read as “x is an
element of A” or “x is in A.”
(2) N is the set of natural numbers: N = {1, 2, 3, . . .}. (We will often consider 0 as a natural number.)
(3) Z is the set of integers: Z = {. . . , −3, −2, −1, 0, 1, 2, . . .}.
(4) Q is the set of rational numbers: Q = { ba : a, b ∈ Z and b 6= 0}.
(5) R is the set of real numbers.
(6) C is the set of complex numbers: C = {a + bi : a, b ∈ R}.
(7) For sets A, B we denote the cartesian product by A × B; we also call this the direct product. The
product of A and B is the set of ordered pairs: A × B = {(a, b) : a ∈ A and b ∈ B}. We often denote
A × A by A2 .
(8) For a set A and a natural number n ∈ N, An is the set of n-tuples of elements in A: An = {(a1 , . . . , an ) :
a1 , . . . , an ∈ A}.
(9) Quantifiers: ∀, ∃ denote the universal (for every) and existential (there exists) quantifiers.
(10) A ⊆ B means A is a subset of B, that is every element of A is an element of B.
(11) A ∩ B means the intersection of A and B: A ∩ B = {x : x ∈ A and x ∈ B}.
(12) A ∪ B means the union of A and B: A ∩ B = {x : x ∈ A or x ∈ B}.
(13) Given a set A, the collection of all subsets of A is a set called the power set and denoted by P(A).
(14) There is a unique set with no elements. This is called the empty set and is denoted by 0.
/
Remark 2. Remember to use your logic skills. The following is a sample of important proof techniques.
(1) Logic of Conditionals.
(a) To prove the statement P → Q is true, first assume that P is true and prove either directly or
indirectly (contradiction) that Q is true.
(b) To prove the bi-conditional P ↔ Q involves both proving the necessity P → Q and sufficiency
Q → P.
(c) Remember that the contrapositive of the statement P → Q is ¬Q → ¬P. A statement and its
contrapositive are tautologically equivalent.
(2) DeMorgan’s Laws: i) ¬(P ∧ Q) is tautologically equivalent to ¬P ∨ ¬Q and ii) ¬(P ∨ Q) is tautologically equivalent to ¬P ∧ ¬Q
(3) The negation of the statement “∀xP(x)” is equivalent to the statement ∃x¬P(x). Similarly, ¬∃xP(x)
is equivalent to ∀x¬P(x).
(4) To prove a universally quantified sentence ∀xP(x) take an arbitrary x and prove that P(x).
(5) Axiom of Extensionality. Two sets A and B are equal if every member of A is a member of B and
every member of B is a member of A. We usually do this in two steps. Take an arbitrary member
of A and explain why in B, then the reverse.
(6) To prove that the conditional P → (Q ∨ R) is true you can start your proof by assuming that P is
true while Q is false. Then you need to show that R is true.
(7) (The Principle of Mathematical Induction) When trying to prove a statement about the natural
numbers use induction: i) Basis Step – prove true for n = 1 ii) Induction Step – assume true for n
(or 1, 2, . . . , n) and prove true for n + 1.
(8) Define by Recursion, prove by induction.
Exercise 1. Let A, B and C be sets. Prove that A ∩ (B ∪C) = (A ∩ B) ∪ (A ∩C).
Exercise 2. Prove that if A is a set with n elements, then P(A) has 2n elements.
2
2. R ELATIONS AND F UNCTIONS
Definition 3. Given sets A and B a relation from A to B is a subset of A × B. We can discuss discuss
different kinds of relations from A to B. The most important class of relations are the functions. Recall that
the relation R from A to B is a function if it satisfies
i. For all a ∈ A there is a b ∈ B such that (a, b) ∈ R.
ii. For any a ∈ A, if b1 , b2 ∈ B such that (a, b1 ), (a, b2 ) ∈ R, then b1 = b2 . [Vertical Line Test.]
We usually write f : A → B and say f is a function from A to B. In this case, A is called the domain of f
and B is called the co-domain.
Relations on A are subsets of A × A.
Exercise 3 (Amy). Supply a definition for the following: i) reflexive relation, ii) symmetric relation, iii)
transitive relation, iv) equivalence relation, v) anti-symmetric relations, vi) partial order.
Exercise 4 (Rachel). Consider the relation on N defined by divisibility:
R = {(a, b) ∈ N2 : ∃k ∈ N such that ak = b}.
Prove that R is a partial order.
Exercise 5 (Rachel, Shan). Supply a definition for a function f : A → B to be: i) injective (one-to-one), ii)
surjective (onto), iii) bijective.
Definition 4. Given sets A, B,C and functions f : A → B and g : B → C, the composition of f and g is the
function g ◦ f : A → C defined by (g ◦ f )(x) = g( f (x)).
Exercise 6. Prove that the composition of two functions is again a function.
Exercise 7 (Harrison). 1. Prove that the composition of two injective functions is injective.
2. Prove that the composition of two surjective functions is surjective.
Definition 5. A bijection from a set onto itself is called a permutation. The set of permutations of the set A
is denoted SA . When A = {1, 2, . . . , n} instead of writing S{1,2,...,n} we write Sn .
Exercise 8. Suppose A and B are two sets with n-elements. Determine the number of bijections from A onto
B.
Exercise 9 (Shan). Determine the size of the set Sn for n ∈ N.
3. O PERATIONS
Definition 6. Given a set A and a n ∈ N, an n-ary operation on A is a function ∗ : An → A. Notice that when
n = 2 we call this a binary operation instead of 2-ary. We also use the terms tertiary (or ternary), unary, and
nullary for n = 3, 1, 0.
For the most part we will be interested in binary operations on sets.
Exercise 10 (Class). How many unary operations on a set with n elements?
Exercise 11 (Class). 1. How many binary operations on a set with 1 element?
2. How many binary operations on a set with 2 elements?
3. How many binary operations on a set with n elements?
Example 7. 1. For any n ∈ N an n-ary operation on A is the function f (a1 , . . . , an ) = a1 .
2. On the sets N and Z two natural binary operations are addition and multiplication.
3. Recall that addition on Q is defined by ba + dc = ad+bc
bd , while multiplication is defined by
ac
bd
=
ac
bd .
3
Definition 8. When we talk about binary operations, say ∗ : A2 → A, instead of writing ∗(a, b) we write
a ∗ b.
The operation ∗ is called associative if for all a, b, c ∈ A, a ∗ (b ∗ c) = (a ∗ b) ∗ c. (Notice that we don’t
write ∗(a, ∗(b, c)) = ∗(∗(a, b), c)... hopefully now you know why we use the notation a ∗ b.)
Exercise 12. [Amy] Which of the following operations on N are associative?
1. a ∗ b = 50.
2. a ∗ b = a.
3. a ∗ b = ab .
4. a ∗ b = pa + qb + r (where p, q, r ∈ N are fixed).
Exercise 13 (Rachel). 1. Define a commutative operation on a set.
2. Suppose A is equipped with two operations, say ∗, ◦. Define what it means for ∗ to distribute over ◦.
Exercise 14. Which of the operations in Example 12 are commutative.
Exercise 15 (Alden). Find all (binary) operations on a set with two elements that are associative. How
about a set with 3 elements? Which of these are commutative?
Exercise 16 (Raymond). Determine whether the cross-product on R3 is associative or commutative.
Exercise 17 (Harrison). Let A be equipped with a binary operation, say +. On A2 the operation (a1 , a2 ) ∗
(b1 , b2 ) = (a1 + a2 , b1 + b2 ). Prove that ∗ is not associative.
Exercise 18 (Alden). Let A be equipped with a binary operation, say +. On A2 we define the coordinatewise + as the operation (a1 , a2 ) ⊕ (b1 , b2 ) = (a1 + b1 , a2 + b2 ). Prove that ⊕ is associative if and only if +
is associative. Similarly, for commutativity.
˙ 1 , s2 ) =
Exercise 19 (Shan). Let A = R × R and consider the operation on A defined by the rule (r1 , r2 )(s
(r1 s1 +r1 s2 +r2 s1 , r2 s2 ). Is this operation associative? Is it commutative? Does it distribute over coordinatewise addition.
˙ 1 , s2 ) =
Exercise 20 (Amy). Let A = R × R and consider the operation on A defined by the rule (r1 , r2 )(s
(r1 s1 , r1 s2 + r2 s1 ). Is this operation associative? Is it commutative? Does it distribute over coordinate-wise
addition.
Exercise 21 (Harrison). Let A be a (nonempty) set and consider the set of all functions from A back into
A. Denote this set by AA . Prove that composition, ◦, is an associative binary operation on AA . Determine
when it is commutative.
˙ 1 , s2 ) = (r1 s1 −
Exercise 22. Let A = R × R and consider the operation on A defined by the rule (r1 , r2 )(s
r1 s2 , r1 s2 + r2 s1 ). Is this operation associative? Is it commutative? Does it distribute over coordinate-wise
addition.
Exercise 23. Let A = R × R and let p be a (fixed) prime. Consider the operation on A defined by the
˙ 1 , s2 ) = (r1 s1 + pr1 s2 , r1 s2 + r2 s1 ). Is this operation associative? Is it commutative? Does it
rule (r1 , r2 )(s
distribute over coordinate-wise addition.
4. S PECIAL E LEMENTS R ELATIVE TO AN O PERATION
Definition 9. Suppose A is a (nonempty) set and ∗ is an operation on A. Let a ∈ A. We say
1. a is a left zero element if for all b ∈ A, a ∗ b = a.
2. a is a right zero element if for all b ∈ A, b ∗ a = a.
3. a is a zero element if it is both a left and right zero element.
4. a is a left identity element if for all b ∈ A, a ∗ b = b.
5. a is a right identity element if for all b ∈ A, a ∗ b = a.
6. a is an identity element if it is both a left identity and right identity.
4
Exercise 24 (Rachel). Suppose (A, ∗) has a, b ∈ A for which a is a left zero and b is a right zero. Prove that
a = b. Conclude that there is at most one zero element for an operation.
Must it be true that a = b if both a and b are left (right) zero elements?
Exercise 25. Suppose (A, ∗) has a, b ∈ A for which a is a left identity and b is a right identity. Prove that
a = b. Conclude that there is at most one identity element for an operation.
Must it be true that a = b if both a and b are left (right) identity elements?
Exercise 26. For the associative operations on the set A = {a, b} identity the left (right) zero, left (right)
identity, if any.
5. S EMIGROUPS AND M ONOIDS
Definition 10. A semigroup is a set A with an associative operation *. We will write “(A, ∗) is a semigroup”.
Definition 11. Suppose (A, ∗) is a semigroup. We recursively define exponentiation on A. For a ∈ A we set
a1 = a, a2 = a ∗ a. Then suppose we have defined an and define an+1 = an ∗ a.
Exercise 27 (Rachel). Let (A, ∗) be a semigroup. Prove that for an element a ∈ A and n ∈ N, that an+1 =
a ∗ an and explain why this might not be true if ∗ is not associative.
Exercise 28 (Raymond). Suppose that (A, ∗) is a semigroup and that ∗ is commutative. Prove that (a∗b)n =
an ∗ bn .
Definition 12. Suppose (A, ∗) is a semigroup. An element a ∈ A for which a2 = a is called an idempotent.
Exercise 29. On N consider the operation gcd, that is, for a, b ∈ N, the operation sends (a, b) to gcd(a, b).
I. Prove that gcd is an associative and commutative operation with a zero element.
II. Prove that every element is idempotent.
Definition 13. Suppose that (A, ∗) is a semigroup. If there is an identity for ∗, then we call (A, ∗) a monoid.
When we say (A, ∗, 1) is a monoid this means two things: i) ∗ is associative and ii) 1 is an identity for ∗.
(Notice that (Z, +, 0) is a monoid. So abstractly we use multiplication as our operation and 1 as its
identity. But in specific examples a different element could be the identity.
Let (A, ∗, 1) be a monoid and a ∈ A. We say a is a left unit if there is some x ∈ A such that a ∗ x = 1. An
element x ∈ A for which a ∗ x = 1 is called a right inverse of a.
Similarly, we define a right unit and a left inverse. An element in a monoid that is both a left unit and
right unit is called a a unit.
Exercise 30 (Alden). Suppose (A, ∗, 1) is a monoid and that a is a unit. Prove that if x is a right inverse
and y is a left inverse, then x = y.
Exercise 31. Suppose (A, ∗, 1) is a monoid and that a is a unit. Prove that if x and y are both right (left)
inverses for a, then x = y.
Exercise 32. In the semigroup (AA , ◦) prove that an element is a left unit if and only if it is surjective.
Exercise 33. In the semigroup (AA , ◦) prove that an element is a right unit if and only if it is injective.
Exercise 34. Find the identity and units in Exercise 19 and Exercise 20.
Example 14. Recall from Matrix Theory that the set of n-matrices over R is denoted by Mn (R), and is
equipped with matrix addition and matrix multiplication. We will assume that both of these operations are
associative. Addition is commutative while multiplication is not if n ≥ 2. The zero matrix is the additive
identity and a zero element with respect to multiplication. The identity matrix is the multiplicative identity.
Exercise 35. Characterize the units of M2 (R).
5
Definition 15. Suppose (A, ∗, 1) is a monoid and that a is a unit. We just proved that an inverse for a is
unique. We denote the inverse of a by a−1 .
Definition 16. Suppose (A, ∗, 1) is a monoid and that a is a unit. Prove that a−1 is a unit. What is (a−1 )−1 ?
Exercise 36. Let (A, ∗, 1) be a monoid and a, b are units of A. Prove that a ∗ b is also a unit. What is its
inverse?
Exercise 37. Let (A, ∗, 1) be a monoid and a a unit of A. Prove that for any n ∈ N, an is a unit. Find (an )−1 .
Exercise 38. Let (A, ∗, 1) be a monoid and a a unit of A. Prove the Law of Exponentiation: for any n, m ∈ Z,
an ∗ am = an+m .
6. LATTICES
We now begin our study of sets equipped with certain kinds of binary operations. In this section we
consider what are known as bounded lattices. The rest of the course will be centered around groups.
Definition 17. Throughout this section L denotes a set that has two associative operations on it ? and •. We
assume that the operations satisfy the following list of axioms.
a. Both ? and • are commutative.
b. Both ? and • have zero elements. Let > denote the zero element for ?, and ⊥ denote the zero
element for •.
c. The operations satisfy what is known as the absorption law: for all a, b ∈ L
[a ? (a • b) = a = a • (a ? b)].
We use “(L, ?, •, >, ⊥)” to say that L is a bounded lattice, that is it satisfies the above axioms.
Exercise 39. Let (L, ?, •, >, ⊥) be a bounded lattice. Prove that i) > is an identity for •, and ii) ⊥ is an
identity for ?.
Exercise 40. Let (L, ?, •, >, ⊥) be a bounded lattice. Consider the relations R = {(a, b) : a ? b = b} and
S = {(a, b) : a • b = a}. Prove that R = S.
Exercise 41. Let (L, ?, •, >, ⊥) be a bounded lattice. Prove that the relation defined in Exercise 40 is a
partial order. Instead of saying that (a, b) ∈ R we write a ≤ b.
Exercise 42. Let (L, ?, •, >, ⊥) be a bounded lattice. Prove that i) for all a ∈ L, a ≤ 1, and ii) for all a ∈ L,
⊥≤ a.
Exercise 43. Let (L, ?, •, >, ⊥) be a bounded lattice. Prove that i) for all a, b ∈ L, a ≤ a ∗ b, and ii) for all
a ∈ L, (b • a) ≤ a.
Exercise 44. Let (L, ?, •, >, ⊥) be a bounded lattice. Prove that i) for all a, b, z ∈ L, if a ≤ z and b ≤ z, then
a ? b ≤ z, and ii) for all a, b, z ∈ L, if z ≤ a and z ≤ b, then z ≤ a • b.
Definition 18. What we have shown is that if (L, ?, •, >, ⊥) is a bounded lattice, then we can equip L with
a partial order ≤ so that (L, ≤) satisfies the properties that for any pair of elements in L there is smallest
element in L which is bigger than both elements. Similarly, for any pair of elements there is a largest
element that is smaller than both elements.
Given a subset S ⊆ L we that the element x ∈ L is an upper bound for S if for all s ∈ S, s ≤ x. If out of
all of the upper bounds for S there exists a particular upper bound, say z, such that z ≤ x for all other upper
bounds x, then z is called a least upper bound for S.
(We analogously defined a lower bound for S and greatest lower bound for S).
In the example of a lattice we know that a, b ≤ a ? b and if a, b ≤ z, then a ? b ≤ z. Thus a ? b is the least
upper bound for a and b.
6
Exercise 45. Suppose S ⊆ L. If z is a greatest lower bound for S, then it is the only greatest lower bound
for S.
Definition 19. Suppose P is a set equipped with a partial order ≤; not necessarily arriving from a lattice structure. Suppose that every pair of elements has a least upper bound and a greatest lower bound.
Furthermore, suppose there are elements 0, 1 ∈ P such that for all a ∈ P, 0 ≤ a and a ≤ 1.
We define binary operations on P: for a, b ∈ P
a ∨ b = l.u.b.{a, b}
and
a ∧ b = g.l.b.{a, b}.
Exercise 46. Prove that ∨ is an associative operation.
Exercise 47. Prove that ∧ is an associative operation.
Exercise 48. Prove that ∨ and ∧ are commutative operations.
Exercise 49. Prove that ∨ and ∧ have zero elements.
Exercise 50. Prove that for all a, b ∈ P a ∨ (a ∧ b) = a ∧ (a ∨ b).
Exercise 51. What is the final conclusion of this section involving partial orders with pairs having a l.u.b.
and g.l.b. and lattices?