1. 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

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

3. 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

4. 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

5. 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

6. 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)

7. 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

8. 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

9. 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

10. 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

11. 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

12. 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

13. 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