ments and a rule of combination, &, satisfying the following axioms: 1. Closure: If g1 E G and g2 E G, then g1 & g2 E G. 2. Associative law: For any g1 g2 and g3 in G, (g1 & g2) & g3 = g1 & (g2 & 3. Identity: There exists an element e in G such that for all g E G, & e = e & g = g. 4. Inverse: For each element g E G there exists an element g such that g &g=e=g&g. 5. Commutative law: If g1 E G and g2 E G, then g1 & g2 = g2 g1. It can be verified that the identity element e is unique as is each element's inverse. The identity element has its own inverse; if h is the inverse of g, then g is the inverse of h. Obviously the integers under ordinary addition are a simple example of a commutative group. All N-dimensional vectors of real numbers form a commutative group under addition. All complex numbers of unit magnitude form a commutative group under multiplication. DEFINITION 5A.2. A subgroup S of the group G is a collection of elements from G which form a group. A subgroup I of G, which is not equal to G, is called a proper subgroup. The identification of a subgroup S within G permits the following proce- dure to be performed. Coset Construction Algorithm: (a) Let Cao = S.n = 0. (b) Construct (c) If Ant1 is empty, then STOP. (d) Increase n by 1. (e) Select an element gn from An. (f) Construct Can={g:g=gn&s,e = S} (g) Go to (b). The sets Ca;, i > 0, are called cosets of the subgroup S. It can be shown that (a) Two elements g and g' are in the same coset if and only if g & Too ES. Pseudonoise Generators (b) There is a one-to-one correspondence between the elements of I and the elements of each coset. (c) Ca; and Ca, are disjoint for i # j, and unique, and hence, when G is finite, (5A.1) where A denotes the number of elements in A, and N - 1 is the number of cosets generated in the above construction algorithm. The number G of ele- ments in a group G is called the order of G. The order of G is prime, then by (5A.), G contains no proper subgroups. Example 5A.1. The integers 0, 1, , 11 under addition modulo 12 form a group