 G. (The term "modulo" is described in Section 5A.2 for those not familiar with it.) One proper subgroup I of G is I = {0, 3, 6,9}, and the cosets of this subgroup are {1, 4, 7, 10} and {2,5,8,11}. Suppose that g is an element of a finite-order commutative group. An ele- ment gm can be constructed by combining m of the elements g according to the rule of combination used in G. Thus, g2=g & g DEFINITION 5A.3. Let e be the identity element of G and let g E G. Then the smallest positive integer k such that (5A.2) is called the exponent of g. It can be shown that: (a) The collection of elements defined by (5A.3) is a subgroup of G. S, g is called the cyclic group generated by g. If there exists a g E G such that S = G, then G is cyclic and g is called a primi- tive element of G. (b) The exponent of g is the order of S . g' (This has led to the interchange- able use of exponent and order in describing an element g.) (c) The exponent of g divided the order of G, i.e., 15°g1 (5A.4) Appendix 5A. Finite Field Arithmetic (The vertical bar between two quantities should be read "divides.") Example 5A.2. Consider the integers relatively prime to 9 under multi- plication mod 9. (The term "relatively prime" is defined in Section 2 for those not familiar with it.) The elements of this collection are 1, 2, 4, 5, 7, and 8. This group has the operation table: A tabulation of the cyclic subgroup generated by each element gives S1 = {1} S2 = {1, 2,4,8,7,5] S4 = {1,4,7} S5 = 4, 2} S7 = {1,7,4} S8 = {1,8} Clearly this six element group is cyclic, the elements 2 and 5 are primitive, 7 and 4 have exponent 3, and 8 has exponent 2. A cyclic group G of order k is isomorphic to the integers 0,1,...,k under addition modulo k. Specifically, using the convention go = e and assum- ing g generates G, the correspondence gm - - m, m=0,1,...,k = , - 1, (5A.5) implies that the group operation tables for G and for modulo k addition are identical. That is, = and only if h = (m + n) mod k, for all m and N. This isomorphism can be used to find the expone