Math 3121 Abstract Algebra I

1. Math 3121Abstract Algebra I Lecture 3 Sections 2-4: Binary Operations, Definition of Group

2. Questions on HW (not to be handed in) • Pages 19-20: 1, 3, 5, 13, 17, 23, 38, 41

3. Section 2: Binary Relations • Definition: A binary relation * on a set S is a function mapping S×S into S. For each (a, b) in S×S, denote *((a, b)) by a * b.

4. Examples • The usual addition for ℤ the integers ℚ the rational numbers ℝ the real numbers ℂ the complex numbers ℤ+ the positive integers ℚ + the positive rational numbers ℝ + the positive real numbers • The usual multiplication for these numbers.

5. 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 ℝ + • division on Z • More?

6. 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.

7. 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.

8. 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)

9. Examples • Composition is associative but not commutative. • Matrix multiplication is associative but not commutative. • More in book.

10. Section 3 • Definition of binary structure • Homomorphism • Isomorphism • Structural properties • Identity elements

11. 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.

12. Homomorphism • Definition (homomorphism of binary structures): Let <S,*> and <S’,*’> be binary structures. A homomorphism from <S,*> to <S’,*’> 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: <S,*>  <S’,*’>.

13. 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.

14. Isomorphism • Definition: A homomorphism of binary structures is called an isomorphism iff the corresponding map of sets is one to one and onto.

15. 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).

16. Identity Element • Definition: Let <S, *> 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

17. Uniqueness of Identity Theorem: A binary structure has at most one identity element. Proof: Let <S, *> 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.

18. Preservation of identity • Theorem: If h: <S, *>  <S’,*’> is an isomorphism of binary algebraic structures, and e is an identity element of <S, *>, then h(e) is an identity element of <S’,*’>. • 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’.

19. Section 4: Groups • Definition: A group <G, *> is a set G together with a binary operation * on G such that • Associatively: For all a, b, c in G (a * b) * c = a * (b * c) • Identity: There is an element e in G such that for all x in G e * x = x * e = x • Inverse: For each x in G, there is an element x’ in G such that x * x’ = x’ * x = e

20. Technicalities • Sometimes use notation <G, *, e, ‘> 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.

21. Examples • <ℤ, +> the integers with addition. • <ℚ, +> the rational numbers with addition. • <ℝ, +> the real numbers with addition. • <ℂ, +> the complex numbers with addition. • The set {-1, 1} under multiplication. • The unit circle <U, ·> 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

22. 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

23. Cancellation Theorem: If G is a group with binary operation *, then left and right cancellation hold a * b = a * c implyb = 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.

24. First Order Equations Theorem: Let <G, *> 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)

25. Uniqueness of Identity and Inverse • Theorem: Let <G, *, e,’> 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)

26. Formula for inverse of product Theorem: Let <G, *, e, ‘> 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)

27. 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