GroupsPhysics and MathematicsSetsMapImageBinary OperationGroup PropertiesDiscrete GroupIsomorphismMatrix RepresentationGroupsGroupsPhysics and MathematicsPhysics and MathematicsClassical theoretical physicists were often the Classical theoretical physicists were often the preeminent mathematicians of their time.preeminent mathematicians of their time.•Descartes – optics; analytic geometryDescartes – optics; analytic geometry•Newton – dynamics, gravity, optics; calculus, algebraNewton – dynamics, gravity, optics; calculus, algebra•Bernoulli – fluids, elasticity; probability and statisticsBernoulli – fluids, elasticity; probability and statistics•Euler – fluids, rotation, astronomy; calculus, geometry, Euler – fluids, rotation, astronomy; calculus, geometry, number theorynumber theory•Lagrange – mechanics, astronomy; calculus, algebra, Lagrange – mechanics, astronomy; calculus, algebra, number theorynumber theory•Laplace – astronomy; probability, differential equationsLaplace – astronomy; probability, differential equations•Hamilton – optics, dynamics; algebra, complex numbersHamilton – optics, dynamics; algebra, complex numbersSetsSetsSet notationSet notation•Set Set XX = { = {xx: : PP((xx)})}Union and intersectionUnion and intersection•XX  Y Y = {= {xx: : xx  XX or or xx  YY}}•XX  Y Y = {= {xx: : xx  XX and and xx  YY}}SubsetSubset•YY  XX ,if ,if  yy  YY, then , then yy  XXCartesian productCartesian product•XX  Y Y = {(= {(x, yx, y)): : xx  XX, , yy  YY}}ABCC = A  BMapMapA map is an association from A map is an association from one set to another.one set to another.•Sets Sets XX = { = {xx}, }, YY = { = {yy}}•Map Map ff: : XX  YY•X X is theis the range range•Y Y is theis the domain domainMaps are also called Maps are also called functions.functions.•ff: : XX  YY or or xx  ff((xx))fx  X, f(x)  YXYImageImageFunctions define subsets Functions define subsets called image sets.called image sets.•ff((XX) = {) = {ff((xx); ); xx  XX}}Injective or one-to-one:Injective or one-to-one:•Any two distinct elements of Any two distinct elements of XX have distinct images in have distinct images in YY.. xx11, , xx22  XX, where , where xx11 ≠≠ xx22, , then then ff((xx11) ) ≠ ≠ ff((xx22).).fXYf(X)Surjective or onto:Surjective or onto:•The image of The image of XX under under ff is the whole of is the whole of YY.. yy  Y,Y,  xx  XX, such that , such that ff((xx) = ) = yy..Binary OperationBinary OperationA binary operation on a set A A binary operation on a set A is a map from is a map from AA  A A  AA..•ff((a,ba,b) = ) = aa ◦◦ bb = = cc; ; a, b, ca, b, c  AAAssociative operation:Associative operation:•aa ◦◦ ((bb ◦◦ cc) = () = (aa ◦◦ bb) ) ◦◦ cc Commutative operation:Commutative operation:•aa ◦◦ bb = = bb ◦◦ aaBinary operations on the real Binary operations on the real numbers numbers RR may be may be associative and associative and commutative.commutative.•Addition is bothAddition is both•Subtraction is neitherSubtraction is neitherMatrix multiplication is Matrix multiplication is associative, but not associative, but not commutative.commutative.S1Group PropertiesGroup PropertiesGroups are sets with a Groups are sets with a binary operation.binary operation.•Call it multiplicationCall it multiplication•Leave out the operator signLeave out the operator signGroup definitions: Group definitions: a, b, c a, b, c  GG•Closure:Closure: ab ab  GG•Associative: Associative: aa((bcbc) = () = (abab))cc•Identity: Identity:  11  GG, , 11aa = = aa1 = 1 = aa, ,  a a  GG•Inverse: Inverse:  aa-1 -1  GG, , aa-1-1aa = = aaaa-1-1 = 1, = 1,  a a  GGProblemProblemAre these subsets of Are these subsets of ZZ, the , the set of integers, groups under set of integers, groups under addition?addition?•ZZ++: {: {nn: : nn  ZZ, , nn > 0} > 0}even numbers: {2even numbers: {2nn: : nn  ZZ}}•odd numbers: {2odd numbers: {2nn+1+1: : nn  ZZ}}•{{±±nn22: : nn  ZZ}}•{0} {0}  { {±±22nn : : nn  ZZ++}}Discrete GroupDiscrete GroupA table can describe a group A table can describe a group with a finite number of with a finite number of elements.elements.Repeated powers of Repeated powers of bb generate all other elements.generate all other elements.•A A cycliccyclic group group•bb is a generator is a generator–bb22 = = cc–bb33 = = dd–bb44 = = aaS1cbadbadcadcbdcbadcbadcbaIsomorphismIsomorphismA group may have other A group may have other ways of realizing the ways of realizing the elements and operation.elements and operation.If the realization is one-to-If the realization is one-to-one and preserves the one and preserves the operation it is isomorphic.operation it is isomorphic.A homomorphism preserves A homomorphism preserves the operation, but is not one-the operation, but is not one-to-one.to-one.S1111111111111iiiiiiiiiiiiThe complex units are isomorphic to the cyclic 4-group.Matrix RepresentationMatrix RepresentationGroups are often Groups are often represented by matrices.represented by matrices.•Unitary matrices with Unitary matrices with determinant 1determinant 1The elements of any finite The elements of any finite group can be represented by group can be represented by unitary matrices.unitary matrices.•Also true for continuous Lie Also true for continuous Lie groupsgroups1001Anext0110B1001C0110DThese matrices are also isomorphic to the cyclic