PRODUCTS OF GROUPS If F and H are groups then their product F x H is the group defined as follows: Example If F = H = {0,1} with + mod 2

PERMUTATIONS A permutation of a set S is a function f : S --> S that is one-to-one and onto The set of all permutation of a set S forms a group, denoted by P(S), under the binary operation, called composition, defined by

EXAMPLES P({a})={(a>a)} has one element P({a,b})={(a>a, b>b), (a>b,b>a)} has two elements P({a,b,c})={(a>a,b>b,c>c), (a>b,b>c,c>a), (a>c,b>a,c>b), (a>a,b>c,c>b), (a>c,b>b,c>a), (a>b,b>a,c>c)} has six elements P({1,…,N}) has N! (N factorial) elements

EXAMPLES The set of rigid transformations that map a geometric object into itself form a group under composition P({a,b,c}) describes the group of rigid transformations of an isosceles triangle P({a,b,c,d}) does not describe the group of rigid transformations of a square

DEFINITIONS A group is abelian if The groups we constructed from numbers are abelian, the permutation and geometric transformation groups are generally not The order of a group is a positive number or infinity that counts its elements An element r in a finite group generates a cyclic subgroup

MORE DEFINITIONS Two groups G and H are isomorphic if there exists a function f: G-->H that is one-to-one and onto and satisfies Here the isomorphism f = (0>1,1>2)

INTERESTING EXAMPLES Chinese Remainder Theorem: with isomorphism 0>(0,0), 1>(1,1), 2>(0,2), 3>(1,0), 4>(0,1), 5>(1,2) Theorem (Fermat): with isomorphism 2>1, 4>2, 3>3, 1>4

SUBGROUPS H is a subgroup of a group G if it is a group under the binary operation on G {0}, {0,2}, has 3 subgroups has 5 subgroups {(0,0)}, {(0,0),{0,1}}, {(0,0}, (0,1)}, {(0,0),(1,0)}, A coset of H in G is a subset of the form

LAGRANGE’s THEOREM The order of a finite group is a multiple of the order of any subgroup Proof. This follows from three facts: 1. G = union of all the cosets of H in G 2. every coset has the same # elements 3. distinct cosets are disjoint To prove 3 assume that

CONSEQUENCES If G is a finite group with order m then If p and q are prime numbers then If p = 3 mod 5 and q = 3 mod 5 then L=5K for some integer K

PUBLIC KEY ENCRYPTION Rivest, Shamir, Adelman (~ to 1978 alg.) 1. You generate huge primes p =3, q=3 mod 5 (by Dirichlet’s Theorem) then distribute N=pq to the public and keep K = ((p-1)(q-1)+1)/5 secret 2. Mr Public generates message then computes & publicly sends encrypted message 3. You decrypt message

PUBLIC KEY ENCRYPTION You are a private encrypter and hold the private key K Mr Public can in theory compute K from the public key N but it will require factorizing N=pq, presumably intractible This algorithm revolutionized secret communications and in particular enabled e-commerce

MIND MENDING EXERCISES Problem 5. Prove all previous assertions Problem 6. Find all subgroups of the group Z of integers and all the cosets of each subgroup Problem 7. Find all subgroups and associated subgroups of S({a,b,c}) Problem 8. Develop a tractible method to compute