1. DEFINITION OF A GROUP A set of elements together with a binary operation that satisfies Associativity: Identity: Inverses:

2. EXAMPLES Sets of integers, rational numbers, real numbers, complex numbers with + Set of vectors (plane/space) with + Sets of nonzero rational numbers, real numbers, complex numbers with x Sets {0,1,…,N-1} with + mod N

3. COUNTEREXAMPLES Sets of integers, rational numbers, real numbers, complex numbers with x Set of space vectors with cross product Set of nonnegative integers with + Sets of integers {0,1,…,N-1} with x modN

4. RESULT Theorem: Inverses are Unique Proof:

5. RESULT Theorem: Proof:

6. RESULT Corollary Set of integers {1,…,N-1} with x modN is a group if and only if N is prime

7. RESULT Corollary Set of integers {1,…,N-1} with x modN is a group if and only if N is prime

8. TABULAR PROOF

9. COMBINATORIAL PROOF Only If Part: If N is not prime then it admits a factorization N=AB where Then and x is not a binary operation on the set {1,...,N-1}

10. COMBINATORIAL PROOF If Part: Assume that It suffices to show that has N-1 elements since then it contains 1. If not then there exists such thatand henceImpossible!

11. EXTENSION Theorem: Ifthen the set forms a group under x mod N. Example: for N=15

12. NEURON FIRING EXERCISES Problem 1 How many ways to compute 2+5+1+4 ? (+ is a binary operation) Problem 2 Validate ex&counterex amples Problem 3 Prove the preceding theorem Problem 3 Build the x table for