# 1 under multiplication forms an abelian group where 1

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) = 1 under multiplication forms an abelian group, where [1 mod n ] is the identity, and if as + nt = 1, then [ s mod n ] is the inverse of [ a mod n ]. Z * n is called the multiplicative group of units modulo n . 2 Example 4.10 Continuing the previous example, let us set n = 15, and enumerate the elements of Z * 15 . They are [1] , [2] , [4] , [7] , [8] , [11] , [13] , [14] . An alternative enumeration is [ ± 1] , [ ± 2] , [ ± 4] , [ ± 7] . 2 Example 4.11 As another special case, consider Z * 5 . We can enumerate the elements of this groups as [1] , [2] , [3] , [4] or alternatively as [ ± 1] , [ ± 2] . 2 20

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Example 4.12 For any positive integer n , the set of n -bit strings under the “exclusive or” operator forms an abelian group, where every bit string is its own inverse. 2 From the above examples, one can see that a group may be infinite or finite. In any case, the order of a group is defined to be the cardinality | G | of the underlying set G defining the group. Example 4.13 The order of Z n is n . 2 Example 4.14 The order of Z * p for prime p is p - 1. 2 Note that in specifying a group, one must specify both the underlying set G as well as the binary operation; however, in practice, the binary operation is often implicit from context, and by abuse of notation, one often refers to G itself as the group. Usually, instead of using a special symbol like ? for an abelian group operator, one instead uses the usual addition (“+”) or multiplication (“ · ”) operators. If an abelian group G is written additively, then the identity element is denoted by 0 G (or just 0 if G is clear from context), and the inverse of an element a G is denoted by - a . For a, b G , a - b denotes a + ( - b ). If n is a positive integer, then n · a denotes a + a + · · · + a , where there are n terms in the sum. Moreover, if n = 0, then n · a denotes 0, and if n is a negative integer then n · a denotes - (( - n ) · a ). If an abelian group G is written multiplicatively, then the identity element is denoted by 1 G (or just 1 if G is clear from context), and the inverse of an element a G is denoted by a - 1 or 1 /a . As usual, one may write ab in place of a · b . For a, b G , a/b denotes a · b - 1 . If n is a positive integer, then a n denotes a · a · · · · · a , where there are n terms in the product. Moreover, if n = 0, then a n denotes 1, and if n is a negative integer, then a n denotes ( a - n ) - 1 . For any particular, concrete abelian group, the most natural choice of notation is clear; however, for a “generic” group, the choice is largely a matter of taste. By convention, whenever we consider a “generic” abelian group, we shall use additive notation for the group operation, unless otherwise specified. We now record a few simple but useful properties of abelian groups. Theorem 4.3 Let G be an abelian group. Then 1. for all a, b, c G , if a + b = a + c , then b = c ; 2. for all a, b G , the equation a + x = b in x has a unique solution in G ; 3. for all a, b G , - ( a + b ) = ( - a ) + ( - b ) ; 4. for all a G , - ( - a ) = a ; 5. for all a G and all n Z , ( - n ) a = - ( na ) = n ( - a ) .
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