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Groups Physics and Mathematics Classical theoretical physicists were often the preeminent mathematicians of their time. • • • • Descartes – optics; analytic geometry Newton – dynamics, gravity, optics; calculus, algebra Bernoulli – fluids, elasticity; probability and statistics Euler – fluids, rotation, astronomy; calculus, geometry, number theory • Lagrange – mechanics, astronomy; calculus, algebra, number theory • Laplace – astronomy; probability, differential equations • Hamilton – optics, dynamics; algebra, complex numbers Sets Set notation • Set X = {x: P(x)} Union and intersection • X Y = {x: x X or x Y} • X Y = {x: x X and x Y} A C Subset B • Y X ,if y Y, then y X Cartesian product • X Y = {(x, y): x X, y Y} C=AB Map A map is an association from one set to another. • • • • Sets X = {x}, Y = {y} Map f: X Y X is the range Y is the domain X f Maps are also called functions. Y • f: X Y or x f(x) x X, f(x) Y Image Functions define subsets called image sets. • f(X) = {f(x); x X} X Injective or one-to-one: • Any two distinct elements of X have distinct images in Y. • x1, x2 X, where x1 ≠ x2, then f(x1) ≠ f(x2). Surjective or onto: • The image of X under f is the whole of Y. • y Y, x X, such that f(x) = y. f f(X) Y 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 A • Addition is both • Subtraction is neither Associative operation: • a ◦ (b ◦ c) = (a ◦ b) ◦ c Commutative operation: • a◦b=b◦a Binary operations on the real S1 numbers R may be associative and commutative. Matrix multiplication is associative, but not commutative. Group Properties Groups are sets with a binary operation. • Call it multiplication • Leave out the operator sign Group definitions: a, b, c G • Closure: ab G • Associative: a(bc) = (ab)c • Identity: 1 G, 1a = a1 = a, aG • Inverse: a-1 G, a-1a = 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: {2n: n Z} odd numbers: {2n+1: n Z} {±n2: n Z} {0} {±2n : n Z+} Discrete Group a b c d a a b c d b c d b c d c d a d a b a b c A table can describe a group S1 with a finite number of elements. Repeated powers of b generate all other elements. • A cyclic group • b is a generator – b2 = c – b3 = d – b4 = a Isomorphism 1 i 1 i 1 1 i 1 i i i 1 i 1 1 1 i 1 i i i 1 i 1 The complex units are isomorphic to the cyclic 4-group. A group may have other S1 ways of realizing the elements and operation. If the realization is one-toone and preserves the operation it is isomorphic. A homomorphism preserves the operation, but is not oneto-one. Matrix Representation Groups are often represented by matrices. • Unitary matrices with determinant 1 The elements of any finite group can be represented by unitary matrices. • Also true for continuous Lie groups 1 0 A 0 1 0 1 B 1 0 1 0 C 0 1 0 1 D 1 0 These matrices are also isomorphic to the cyclic 4-group. next