Slide 1Slide 2Slide 3Slide 4Slide 5Slide 6Slide 7Slide 8Slide 9Slide 10Slide 11Slide 12Slide 13Slide 14Slide 15Slide 16Slide 17Slide 18To check “group-ness”Some examples…Slide 21Slide 22Slide 23Slide 24Slide 25Slide 26Slide 27And some properties…Slide 29Slide 30Slide 31Slide 32Slide 33Slide 34Slide 35Slide 36Slide 37Slide 38Slide 39Slide 40SubgroupsExamplesSlide 43Quick facts (identity)Quick facts (inverse)Lagrange’s TheoremSlide 47Slide 48Slide 49Slide 50Slide 51Slide 52Slide 53Slide 54Slide 5515-251Great Theoretical Ideas in Computer ScienceAlgebraic Structures: Group TheoryLecture 15 (October 14, 2008)Today we are going to study the abstract properties of binary operationsRotating a Square in SpaceImagine we can pick up the square, rotate it in any way we want, and then put it back on the white frameIn how many different ways can we put the square back on the frame?R90R180R270R0F|F—F FWe will now study these 8 motions, called symmetries of the squareSymmetries of the SquareYSQ = { R0, R90, R180, R270, F|, F—, F , F }CompositionDefine the operation “-” to mean “first do one symmetry, and then do the next”For example,R90 - R180Question: if a,b YSQ, does a - b YSQ? Yes!means “first rotate 90˚ clockwise and then 180˚”= R270F| - R90means “first flip horizontally and then rotate 90˚”= FR90R180R270R0F|F—F FR0R90R180R270F|F—FFR0R90R180R270F|F—F FR90R180R270F|F—FFR180R270R0R270R0R90R0R90R180F F F|F—F—F|F FF F F—F|F F—FF F|FF—F F|F|F F—R0R0R0R0R180R90R270R180R270R90R270R90R180R90R270R180Some FormalismIf S is a set, S S is:the set of all (ordered) pairs of elements of SS S = { (a,b) | a S and b S }If S has n elements, how many elements does S S have? n2Formally, - is a function from YSQ YSQ to YSQ - : YSQ YSQ → YSQAs shorthand, we write -(a,b) as “a - b”“-” is called a binary operation on YSQDefinition: A binary operation on a set S is a function : S S → SExample:The function f: → defined byis a binary operation on f(x,y) = xy + yBinary OperationsIs 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) AssociativityExamples:Is f: → defined by f(x,y) = xy + yassociative?(ab + b)c + c = a(bc + c) + (bc + c)?NO!YES!A binary operation on a set S is commutative ifFor all a,bS, a b = b a CommutativityIs the operation - on the set of symmetries of the square commutative? NO!R90 - F| ≠ F| - R90R0 is like a null motionIs this true: a YSQ, a - R0 = R0 - a = a?R0 is called the identity of - on YSQIn 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 SIdentitiesYES!InversesDefinition: The inverse of an element a YSQ is an element b such that:a - b = b - a = R0 Examples:R90inverse: R270 R180inverse: R180 F|inverse: F|Every element in YSQ has a unique inverseR90R180R270R0F|F—F FR0R90R180R270F|F—FFR0R90R180R270F|F—F FR90R180R270F|F—FFR180R270R0R270R0R90R0R90R180F F F|F—F—F|F FF F F—F|F F—FF F|FF—F F|F|F F—R0R0R0R0R180R90R270R180R270R90R270R90R180R90R270R1803. (Inverses) For every a S there is b S such that:GroupsA group G is a pair (S,), where S is a set and is a binary operation on S such that:1. is associative2. (Identity) There exists an element e S such that:e a = a e = a, for all a S a b = b a = eCommutative or “Abelian” Groupsremember, “commutative” meansa b = b a for all a, b in SIf G = (S,) and is commutative, then G is called a commutative groupTo 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 identity4. Check every element has an inverseSome examples…ExamplesIs (,+) a group? Is + associative on ?YES!Is there an identity? YES: 0Does every element have an inverse?NO!(,+) is NOT a groupExamplesIs (Z,+) a group? Is + associative on Z?YES!Is there an identity? YES: 0Does every element have an inverse?YES!(Z,+) is a groupExamplesIs (Odds,+) a group? (Odds,+) is NOT a groupIs + associative on Odds?YES!Is there an identity? YES: 0Does every element have an inverse?YES!Are the Odds closed under addition NO!ExamplesIs (YSQ, -) a group? Is - associative on YSQ?YES!Is there an identity? YES: R0Does every element have an inverse?YES!(YSQ, -) is a groupExamplesIs (Zn,+) a group? Is + associative on Zn?YES!Is there an identity? YES: 0Does every element have an inverse?YES!(Zn, +) is a group(Zn is the set of integers modulo n)ExamplesIs (Zn,*) a group? Is * associative on Zn?YES!Is there an identity? YES: 1Does every element have an inverse?NO!(Zn, *) is NOT a group(Zn is the set of integers modulo n)ExamplesIs (Zn*, *) a group? Is * associative on Zn* ?YES!Is there an identity? YES: 0Does every element have an inverse?YES!(Zn*, *) is a group(Zn* is the set of integers modulo nthat are relatively prime to n)And some properties…Theorem: A group has at most one identity elementProof:Suppose e and f are both identities of G=(S,)Then f = e f = eIdentity Is UniqueWe denote this identity by “e”Theorem: Every element in a group has a unique inverseProof:Inverses Are UniqueSuppose b and c are both inverses of a Then b = b e = b (a c) = (b a) c = cOrders and generatorsA group G=(S,) is finite if S is a finite setDefine |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 = eefe feffeOrder of a groupGeneratorsA 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 TQuestion: Does {R90} generate YSQ?Question: Does {F|, R90} generate YSQ?An element g S is called a generator of G=(S,) if {g} generates GDoes YSQ have a generator?NO!YES!NO!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 nProof (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 happenIf
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