1. Groups

2. Physics and Mathematics  Classical theoretical physicists were often the 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, number theoryEuler – fluids, rotation, astronomy; calculus, geometry, number theory Lagrange – mechanics, astronomy; calculus, algebra, number theoryLagrange – mechanics, astronomy; calculus, algebra, number theory Laplace – astronomy; probability, differential equationsLaplace – astronomy; probability, differential equations Hamilton – optics, dynamics; algebra, complex numbersHamilton – optics, dynamics; algebra, complex numbers

3. Sets  Set notation Set X = { x : P ( x )}Set X = { x : P ( x )}  Union and intersection X  Y = { x : x  X or x  Y}X  Y = { x : x  X or x  Y} X  Y = { x : x  X and x  Y}X  Y = { x : x  X and x  Y}  Subset Y  X,if  y  Y, then y  XY  X,if  y  Y, then y  X  Cartesian product X  Y = {( x, y) : x  X, y  Y}X  Y = {( x, y) : x  X, y  Y} A B C C = A  B

4. Map  A map is an association from one set to another. Sets X = { x }, Y = { y }Sets X = { x }, Y = { y } Map f: X  YMap f: X  Y X is the rangeX is the range Y is the domainY is the domain  Maps are also called functions. f: X  Y or x  f( x )f: X  Y or x  f( x ) f x  X, f( x )  Y X Y

5. Image  Functions define subsets called image sets. f(X) = {f( x ); x  X}f(X) = {f( x ); x  X}  Injective or one-to-one: Any two distinct elements of X have distinct images in Y.Any two distinct elements of X have distinct images in Y.  x 1, x 2  X, where x 1 ≠ x 2, then f( x 1 ) ≠ f( x 2 ).  x 1, x 2  X, where x 1 ≠ x 2, then f( x 1 ) ≠ f( x 2 ). f X Y f(X)f(X)  Surjective or onto: The image of X under f is the whole of Y.The image of X under f is the whole of Y.  y  Y,  x  X, such that f( x ) = y.  y  Y,  x  X, such that f( x ) = y.

6. Binary Operation  A binary operation on a set A is a map from A  A  A. f (a,b ) = a ◦ b = c ; a, b, c  Af (a,b ) = a ◦ b = c ; a, b, c  A  Associative operation: a ◦ ( b ◦ c ) = ( a ◦ b ) ◦ ca ◦ ( b ◦ c ) = ( a ◦ b ) ◦ c  Commutative operation: a ◦ b = b ◦ aa ◦ b = b ◦ a  Binary operations on the real numbers R may be associative and commutative. Addition is both Subtraction is neither  Matrix multiplication is associative, but not commutative. S1S1

7. Group Properties  Groups are sets with a binary operation. Call it multiplicationCall it multiplication Leave out the operator signLeave out the operator sign  Group definitions: a, b, c  G Closure: ab  GClosure: ab  G Associative: a ( bc ) = ( ab ) cAssociative: a ( bc ) = ( ab ) c Identity:  1  G, 1a = a1 = a,  a  GIdentity:  1  G, 1a = a1 = a,  a  G Inverse:  a -1  G, a -1 a = aa -1 = 1,  a  GInverse:  a -1  G, a -1 a = aa -1 = 1,  a  G Problem  Are these subsets of Z, the set of integers, groups under addition? Z + : { n : n  Z, n > 0} even numbers: {2 n : n  Z } odd numbers: {2 n+1 : n  Z } {± n 2 : n  Z } {0}  {± 2 n : n  Z + }

8. Discrete Group  A table can describe a group with a finite number of elements.  Repeated powers of b generate all other elements. A cyclic group b is a generator –b 2 = c –b 3 = d –b 4 = a S1S1

9. Isomorphism  A group may have other ways of realizing the elements and operation.  If the realization is one-to- one and preserves the operation it is isomorphic.  A homomorphism preserves the operation, but is not one- to-one. S1S1 The complex units are isomorphic to the cyclic 4-group.

10. Matrix Representation  Groups are often represented by matrices. Unitary matrices with determinant 1Unitary matrices with determinant 1  The elements of any finite group can be represented by unitary matrices. Also true for continuous Lie groupsAlso true for continuous Lie groups next These matrices are also isomorphic to the cyclic 4-group.