Math 3121 Abstract Algebra I Lecture 3 Sections 2-4: Binary Operations, Definition of Group Questions on HW (not to be handed in) Pages 19-20: 1, 3, 5, 13, 17, 23, 38, 41 Section 2: Binary Relations

Definition: A binary relation * on a set S is a function mapping SS into S. For each (a, b) in SS, denote *((a, b)) by a * b. Examples The usual addition for the integers the rational numbers the real numbers

the complex numbers + the positive integers + the positive rational numbers + the positive real numbers The usual multiplication for these numbers. Counterexamples If * is a binary operation on a S, then a*b must

be defined for all a, b in S. Examples of where it is not: subtraction on + division on Z More? More examples Matrices Addition is a binary operation on real n by m

matrices. What about multiplication? Functions For real valued functions of a real variable, addition, multiplication, subtraction, and composition are all binary operations. Closure

Suppose * is a binary operation on a set S, and H is a subset of S. The subset H is closed under *, iff a*b is in H for all a, b in H. In that case the binary operation on H given by restricting * to members of H is called the induced operation of * on H. Examples in book: squares under addition and multiplication of positive integers.

Commutative and Associative Definition: A binary operation * on a set S is commutative iff, for all a, b in S a*b=b*a Definition: A binary operation * on a set S is associative iff, for all a, b, c in S (a * b) * c = a * (b * c)

Examples Composition is associative but not commutative. Matrix multiplication is associative but not commutative. More in book. Section 3

Definition of binary structure Homomorphism Isomorphism Structural properties

Identity elements Binary Structures Definition (Binary algebraic structure): A binary algebraic structure is a set together with a binary operation on it. This is denoted by an ordered pair < S, *> in which S is a set and * is a binary operation on S.

Homomorphism Definition (homomorphism of binary structures): Let ~~ and ~~~~ be binary structures. A homomorphism from ~~~~ to ~~~~ is a map h: S S that satisfies, for all x, y in S: h(x*y) = h(x)*h(y) We can denote it by h: ~~~~ ~~~~. ~~

Examples Let f(x) = ex. Then f is a homomorphism from the real numbers with addition to the positive real numbers with multiplication. Let g(x) = eix. Then g is a homomorphism from the real numbers with addition to unit circle in the complex plane. Isomorphism

Definition: A homomorphism of binary structures is called an isomorphism iff the corresponding map of sets is one to one and onto. Examples Let f(x) = ex. Then f is an isomorphism from the real numbers with addition to the positive real numbers with multiplication.

Let g(x) = eix. Then g is not an isomorphism from the real numbers with addition to unit circle in the complex plane. (not 1-1). Identity Element Definition: Let ~~ be a binary structure. An element e of S is an identity element for * iff , for all x in S: e*x=x*e=x~~

Uniqueness of Identity Theorem: A binary structure has at most one identity element. Proof: Let ~~ be a binary structure. If e1 and e2 are both identities, then e1* e2 = e1 because e2 is an identity and e1* e2 = e2 because e1 is an identity. ~~

Thus e1 = e2. Preservation of identity Theorem: If h: ~~ ~~~~ is an isomorphism of binary algebraic structures, and e is an identity element of ~~~~, then h(e) is an identity element of ~~~~. Proof: Let e be the identity element of S. For each x in S. There is an x in S such that h(x) = x. Then~~

h(x * e) = h (e * x) = h(x). By the homomorphism property, h(x) * h(e) = h (e) * h(x) = h(x). Thus x * h(e) = h(e) * x = x. Thus h(e) is the identity of S. Section 4: Groups Definition: A group is a set G together with a binary operation * on G such that 1) Associatively: For all a, b, c in G (a * b) * c = a * (b * c)

2) Identity: There is an element e in G such that for all x in G e*x=x*e=x 3) Inverse: For each x in G, there is an element x in G such that x * x = x * x = e

Technicalities Sometimes use notation to denote all its components. Specifying the inverse of each element x of G is a unary operation on the set G. Specifying the identity call be considered an operation with no arguments (n-ary where n = zero). We often drop the <> notation and use the set to denote the group when the binary operation is

understood. We already drop the e, and . Later we will show they are uniquely determined. Examples

<, +> the integers with addition. <, +> the rational numbers with addition. <, +> the real numbers with addition. <, +> the complex numbers with addition. The set {-1, 1} under multiplication. The unit circle __ in the complex plane under multiplication.__

Many more in the book positive rationals and reals under multiplication nonzero rationals and reals under multiplication N by M real matrices under addition Elementary Properties of Groups Cancellation: Left: a*b = a*c implies b = c Right: b*a = c*a implies b = c

Unique solutions of a*x = b Only one identity Formula for inverse of product Cancellation Theorem: If G is a group with binary operation *, then left and right cancellation hold a * b = a * c imply b = c b * a = c * a imply b = c

Proof: Suppose a * b = a * c. Then there is an inverse a to a. Apply this inverse on the left: a * (a * b) = a *(a * c) (a * a ) * b = (a * a) * c associatively e * b = e * c inverse b = c identity Similarly for right cancellation.

First Order Equations Theorem: Let be a group. If a and b are in G, then a*x = b has a unique solution and so does x*a = b. Proof: (in class solve for x by applying inverse, uniqueness follows by cancellation) Uniqueness of Identity and Inverse

Theorem: Let be a group. 1) There is only one element y in G such that y * x = x * y = x, for all x in G, and that element is e. 2) For each x in G, there is only one element y such that y*x = x*y = e, and that element is x. Proof: 1) is already true for binary algebraic structures. 2) proof in class (use cancellation) Formula for inverse of product

Theorem: Let be a group. For all, a, b in G, the inverse is given by (a * b) = b*a. Proof: Show it gives and inverse (in class) HW (due Tues, Oct 7) Not to hand in: Pages 45-49: 1, 3, 5, 21, 25 Hand in: pages 45-49: 2, 19, 24, 31, 35