lecture16.ppt
- 1. 15-251 Great Theoretical Ideas in Computer Science
- 2. Algebraic Structures: Group Theory Lecture 16, October 14, 2009
- 3. Today we are going to study the abstract properties of binary operations
- 4. Rotating a Square in Space Imagine we can pick up the square, rotate it in any way we want, and then put it back on the white frame
- 5. In how many different ways can we put the square back on the frame? R90 R180 R270 R0 F| F— F F We will now study these 8 motions, called symmetries of the square
- 6. Symmetries of the Square YSQ = { R0, R90, R180, R270, F|, F—, F , F }
- 7. Composition Define the operation “” to mean “first do one symmetry, and then do the next” For example, R90 R180 Question: if a,b YSQ, does a b YSQ? means “first rotate 90˚ clockwise and then 180˚” = R270 F| R90 means “first flip horizontally and then rotate 90˚” = F
- 8. R90 R180 R270 R0 F| F— F F R0 R90 R180 R270 F| F— F F R0 R90 R180 R270 F| F— F F R90 R180 R270 F| F— F F R180 R270 R0 R270 R0 R90 R0 R90 R180 F F F| F— F— F| F F F F F— F| F F— F F F| F F— F F| F| F F— R0 R0 R0 R0 R180 R90 R270 R180 R270 R90 R270 R90 R180 R90 R270 R180
- 9. Some Formalism If S is a set, S S is: the set of all (ordered) pairs of elements of S S S = { (a,b) | a S and b S } If S has n elements, how many elements does S S have? n2 Formally, is a function from YSQ YSQ to YSQ : YSQ YSQ → YSQ As shorthand, we write (a,b) as “a b”
- 10. “” is called a binary operation on YSQ Definition: A binary operation on a set S is a function : S S → S Example: The function f: → defined by is a binary operation on f(x,y) = xy + y Binary Operations
- 11. Is the operation on the set of symmetries of the square associative? A binary operation on a set S is associative if: for all a,b,cS, (ab)c = a(bc) Associativity Examples: Is f: → defined by f(x,y) = xy + y associative? (ab + b)c + c = a(bc + c) + (bc + c)? NO! YES!
- 12. A binary operation on a set S is commutative if For all a,bS, a b = b a Commutativity Is the operation on the set of symmetries of the square commutative? NO! R90 F| ≠ F| R90
- 13. R0 is like a null motion Is this true: a YSQ, a R0 = R0 a = a? R0 is called the identity of on YSQ In general, for any binary operation on a set S, an element e S such that for all a S, e a = a e = a is called an identity of on S Identities YES!
- 14. Inverses Definition: The inverse of an element a YSQ is an element b such that: a b = b a = R0 Examples: R90 inverse: R270 R180 inverse: R180 F| inverse: F|
- 15. Every element in YSQ has a unique inverse
- 16. R90 R180 R270 R0 F| F— F F R0 R90 R180 R270 F| F— F F R0 R90 R180 R270 F| F— F F R90 R180 R270 F| F— F F R180 R270 R0 R270 R0 R90 R0 R90 R180 F F F| F— F— F| F F F F F— F| F F— F F F| F F— F F| F| F F— R0 R0 R0 R0 R180 R90 R270 R180 R270 R90 R270 R90 R180 R90 R270 R180
- 17. 3. (Inverses) For every a S there is b S such that: Groups A group G is a pair (S,), where S is a set and is a binary operation on S such that: 1. is associative 2. (Identity) There exists an element e S such that: e a = a e = a, for all a S a b = b a = e
- 18. Commutative or “Abelian” Groups remember, “commutative” means a b = b a for all a, b in S If G = (S,) and is commutative, then G is called a commutative group
- 19. To check “group-ness” Given (S,) 1. Check “closure” for (S,) (i.e, for any a, b in S, check a b also in S). 2. Check that associativity holds. 3. Check there is a identity 4. Check every element has an inverse
- 20. Some examples…
- 21. Examples Is (,+) a group? Is + associative on ? YES! Is there an identity? YES: 0 Does every element have an inverse? NO! (,+) is NOT a group
- 22. Examples Is (Z,+) a group? Is + associative on Z? YES! Is there an identity? YES: 0 Does every element have an inverse? YES! (Z,+) is a group
- 23. Examples Is (Odds,+) a group? (Odds,+) is NOT a group Is + associative on Odds? YES! Is there an identity? YES: 0 Does every element have an inverse? YES! Are the Odds closed under addition NO!
- 24. Examples Is (YSQ, ) a group? Is associative on YSQ? YES! Is there an identity? YES: R0 Does every element have an inverse? YES! (YSQ, ) is a group
- 25. Examples Is (Zn,+) a group? Is + associative on Zn? YES! Is there an identity? YES: 0 Does every element have an inverse? YES! (Zn, +) is a group (Zn is the set of integers modulo n)
- 26. Examples Is (Zn,*) a group? Is * associative on Zn? YES! Is there an identity? YES: 1 Does every element have an inverse? NO! (Zn, *) is NOT a group (Zn is the set of integers modulo n)
- 27. Examples Is (Zn*, *) a group? Is * associative on Zn* ? YES! Is there an identity? YES: 1 Does every element have an inverse? YES! (Zn*, *) is a group (Zn* is the set of integers modulo n that are relatively prime to n)
- 29. Theorem: A group has at most one identity element Proof: Suppose e and f are both identities of G=(S,) Then f = e f = e Identity Is Unique We denote this identity by “e”
- 30. Theorem: Every element in a group has a unique inverse Proof: Inverses Are Unique Suppose b and c are both inverses of a Then b = b e = b (a c) = (b a) c = c
- 32. A group G=(S,) is finite if S is a finite set Define |G| = |S| to be the order of the group (i.e. the number of elements in the group) What is the group with the least number of elements? How many groups of order 2 are there? G = ({e},) where e e = e e f e f e f f e Order of a group
- 33. Generators A set T S is said to generate the group G = (S,) if every element of S can be expressed as a finite product of elements in T Question: Does {R90} generate YSQ? Question: Does {F|, R90} generate YSQ? An element g S is called a generator of G=(S,) if {g} generates G Does YSQ have a generator? NO! YES! NO!
- 34. Generators For (Zn,+) Any a Zn such that GCD(a,n)=1 generates (Zn,+) Claim: If GCD(a,n) =1, then the numbers a, 2a, …, (n-1)a, na are all distinct modulo n Proof (by contradiction): Suppose xa = ya (mod n) for x,y {1,…,n} and x ≠ y Then n | a(x-y) Since GCD(a,n) = 1, then n | (x-y), which cannot happen
- 35. If G = (S,), we use an denote (a a … a) n times Definition: The order of an element a of G is the smallest positive integer n such that an = e The order of an element can be infinite! Example: The order of 1 in the group (Z,+) is infinite What is the order of F| in YSQ? 2 What is the order of R90 in YSQ? 4 Order of an element
- 36. Theorem: If G is a finite group, then for g in G, order(g) is finite. Orders For (Zn, +), recall that order(g) = n/GCD(n,g)
- 37. What about (Zn *, *)? Orders order(Zn *, *) = Á(n) What about the order of its elements?
- 38. What about (Zn *, *)? Orders order(Zn *, *) = Á(n) What about the order of its elements? Non-trivial theorem: There are Á(n-1) generators of (Zn*, *)
- 39. Theorem: Let x be an element of G. The order of x divides the order of G Orders Corollary: If p is prime, ap-1 = 1 (mod p) (remember, this is Fermat’s Little Theorem) BTW, what group did we apply the theorem to? G = (Zp*, *), order(G) = p-1
- 41. Subgroups Suppose G = (S,) is a group. If T µ S, and if H = (T, ) is also a group, then H is called a subgroup of G.
- 42. Examples (Z, +) is a group and (Evens, +) is a subgroup. Also, (Z, +) is a subgroup of (Z, +). (Duh!) What about (Odds, +)?
- 43. Examples (Zn, +n) is a group and if k | n, what about ({0, k, 2k, 3k, …, (n/k-1)k}, +n) ? Is (Zk, +k) a subgroup of (Zn, +n)? Is (Zk, +n) a subgroup of (Zn, +n)?
- 44. Quick facts (identity) If e is the identity in G = (S,), what is the identity in H = (T,)?
- 45. Quick facts (inverse) If b is a’s inverse in G = (S,), what is a’s inverse in H = (T,)?
- 46. Lagrange’s Theorem Theorem: If G is a finite group, and H is a subgroup then the order of H divides the order of G. In symbols, |H| divides |G|. Corollary: If x in G, then order(x) divides |G|. Proof of Corollary: Consider the set Tx = (x, x2 = x x, x3, …) H = (Tx, ) is a group. (check!) Hence it is a subgroup of G = (S, ). Order(H) = order(x). (check!)
- 47. On to other algebraic definitions
- 48. We often define more than one operation on a set For example, in Zn we can do both addition and multiplication modulo n A ring is a set together with two operations Lord Of The Rings
- 49. Definition: A ring R is a set together with two binary operations + and ×, satisfying the following properties: 1. (R,+) is a commutative group 2. × is associative 3. The distributive laws hold in R: (a + b) × c = (a × c) + (b × c) c × (a + b) = (c × a) + (c × b)
- 50. Examples: Is (Z, +, *) a ring? How about (Z, +, min)?
- 51. Ring Unit Ring (mult. identity) Division Ring (mult. identity, mult. inverse) Commutative Ring (mult. is commutative) Field (mult. identity, mult. inverse, mult. is commutative)
- 52. Definition: A field F is a set together with two binary operations + and ×, satisfying the following properties: 1. (F,+) is a commutative group 2. (F-{0},×) is a commutative group 3. The distributive law holds in F: (a + b) × c = (a × c) + (b × c) Fields
- 53. Examples: Is (Z, +, *) a field? How about (R, +, *)? How about (Zn, +n, *n)?
- 54. Why should I care about any of this? Groups, Rings and Fields are examples of the principle of abstraction: the particulars of the objects are abstracted into a few simple properties If you prove results from some group, check if the results carry over to any group In The End…
- 55. Symmetries of the Square Compositions Groups Binary Operation Identity and Inverses Basic Facts: Inverses Are Unique Generators Rings and Fields Definition Here’s What You Need to Know…