10/12/2010115-251Great Theoretical Ideas in Computer ScienceAlgebraic Structures: Group TheoryLecture 15 (October 12, 2010)Number TheoryNaturalsclosed under +a+b = b+aa+0 = 0+aIntegersclosed under +a+b = b+aa+0 = 0+aa+(-a) = 0Znclosed under +na+nb = b+naa+n0 = 0+naa+n(-a) = 0(a+b)+c = a+(b+c)(a+b)+c = a+(b+c)(a+nb)+nc = a+n(b+nc)A+0 = 0+Aa+0 = 0+aa+(-a) = 0a+n0 = 0+naa+n(-a) = 0closed under *closed under *nclosed under *(a+b)*c = a*c+b*cdittodittoA+(-A) = 01/a may not existdittodittoNumber Theoryclosed under +A+B = B+Aclosed under +a+b = b+aclosed under +na+nb = b+na(a+b)+c = a+(b+c)(A+B)+C = A+(B+C)MatricesIntegersZn(a+nb)+nc = a+n(b+nc)A+0 = 0+Aa+0 = 0+aa+(-a) = 0a+n0 = 0+naa+n(-a) = 0closed under *closed under *nclosed under *(a+b)*c = a*c+b*cdittodittoA+(-A) = 01/a exists if a ≠ 0dittodittoNumber Theoryclosed under +A+B = B+Aclosed under +a+b = b+aclosed under +na+nb = b+na(a+b)+c = a+(b+c)(a+nb)+nc = a+n(b+nc)(A+B)+C = A+(B+C)Invertible MatricesRationalsZn(n prime) Abstraction: Abstract away the inessential features of a problem====10/12/20102Today 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—R0R0R0R0R180R90R270R180R270R90R270R90R180R90R270R18010/12/20103How many symmetries for n-sided body? 2nR0, R1, R2, …, Rn-1F0, F1, F2, …, Fn-1RiRj= Ri+jRiFj= Fj-iFjRi= Fj+iFiFj= Rj-iSome 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× YSQto 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: N × N → N defined byis a binary operation on Nf(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,c∈S, (a♦b)♦c = a♦(b♦c)AssociativityExamples:Is f: N × N → N 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,b∈S, a ♦ b = b ♦ aCommutativityIs the operation • on the set of symmetries of the square commutative? NO!R90• F|≠ F|• R90R0is like a null motionIs this true: ∀a ∈ YSQ, a • R0= R0 • a = a?R0is 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!10/12/20104InversesDefinition: The inverse of an element a ∈ YSQis an element b such that:a • b = b • a = R0Examples:R90inverse: R270R180inverse: R180F|inverse: F|Every element in YSQhas 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 ∈ Sa ♦ 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 inverse10/12/20105Some examples…ExamplesIs (N,+) a group?Is + associative on N?YES!Is there an identity? YES: 0Does every element have an inverse? NO!(N,+) is NOT a groupIs N closed under +?YES!ExamplesIs (Z,+) a group?Is + associative on Z?YES!Is there an identity? YES: 0Does every element have an inverse? YES!(Z,+) is a groupIs Z closed under +?YES!ExamplesIs (Odds,+) a group?(Odds,+) is NOT a groupIs + associative on Odds?YES!Is there an identity? NO!Does every element have an inverse? YES!Is Odds closed under +?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 groupthe “dihedral” group D4Is YSQclosed under •?YES!ExamplesIs (Zn,+n) a group?Is +nassociative on Zn?YES!Is there an identity? YES: 0Does every element have an inverse? YES!(Zn, +n) is a group(Znis the set of integers modulo n)Is Znclosed under +n?YES!10/12/20106ExamplesIs (Zn,*n) a group?Is *nassociative on Zn?YES!Is there an identity? YES: 1Does every element have an inverse? NO!(Zn, *n) is NOT a group(Znis the set of integers modulo n)ExamplesIs (Zn*, *n) a group?Is *nassociative on Zn* ?YES!Is there an identity? YES: 1Does every element have an inverse?YES!(Zn*, *n) is a group(Zn* is the set of integers modulo nthat are relatively prime to n)(Z, *)(Q, *)the rationals(Q \ {0}, *)No inverses…Zero has no inverse…Yes3. (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 ∈ Sa ♦ b = b ♦ a = eSome properties of groups…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”⇒ exactly one
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