# 18 chapter 4 abelian groups this chapter reviews the

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Chapter 4 Abelian Groups This chapter reviews the notion of an abelian group. 4.1 Definitions, Basic Properties, and Some Examples Definition 4.1 An abelian group is a set G together with a binary operation ? on G such that 1. for all a, b G , a ? b = b ? a (commutivity property), 2. for all a, b, c G , a ? ( b ? c ) = ( a ? b ) ? c (associativity property), 3. there exists e G (called the identity element ) such that for all a G , a ? e = a (identity property), 4. for all a G there exists a 0 G such that a ? a 0 = e (inverse property). Before looking at examples, let us state some very basic properties of abelian groups that follow directly from the definition. Theorem 4.2 Let G be an abelian group with operator ? . Then we have 1. the identity element is unique, i.e., there is only one element e G such that a ? e = a for all a G ; 2. inverses are unique, i.e., for all a G , there is only one element a 0 G such that a ? a 0 is the identity. Proof. Suppose e, e 0 are identities. Then since e is an identity, by the identity property in the definition, we have e 0 ? e = e 0 . Similarly, since e 0 is an identity, we have e ? e 0 = e . By the commutivity property, we have e ? e 0 = e 0 ? e . Thus, e 0 = e 0 ? e = e, and so we see that there is only one identity. Now let a G , and suppose that a ? a 0 = e and a ? a 00 = e . Now, a ? a 0 = e implies a 00 ? ( a ? a 0 ) = a 00 ? e . Using the associativity and commutivity properties, the left-hand side can be written a 0 ? ( a ? a 00 ), and by the identity property, the right-hand side can be written a 00 . Thus, we have 19
a 0 ? ( a ? a 00 ) = a 00 . This, together with the equation a ? a 00 = e implies that a 0 ? e = a 00 , and again applying the identity property, we have a 0 = a 00 . That proves a has only one inverse. 2 The above proof was very straightforward, yet quite tedious if one fills in all the details. In the sequel, we shall leave proofs of this type as exercises for the reader. There are many examples of abelian groups. Example 4.1 The set of integers Z under addition forms an abelian group, with 0 being the identity, and - a being the inverse of a Z . 2 Example 4.2 For integer n , the set n Z = { nz : z Z } under addition forms as abelian group, again, with 0 being the identity, and n ( - z ) being the inverse of nz . 2 Example 4.3 The set of non-negative integers under addition does not form an abelian group, since inverses do not exist for integers other than 0. 2 Example 4.4 The set of integers under multiplication does not form an abelian group, since inverses do not exist for integers other than ± 1. 2 Example 4.5 The set of integers 1 } under multiplication forms an abelian group, with 1 being the identity, and - 1 is its own inverse. 2 Example 4.6 The set of rational numbers Q = { a/b : a, b Z , b 6 = 0 } under addition forms an abelian group, with 0 being the identity, and ( - a ) /b being the inverse of a/b . 2 Example 4.7 The set of non-zero rational numbers Q * under multiplication forms a group, with 1 being the identity, and b/a being the inverse of a/b . 2 Example 4.8 The set Z n under addition forms an abelian group, where [0 mod n ] is the identity, and where [ - a mod n ] is the inverse of [ a mod n ]. 2 Example 4.9 The set Z * n of residue classes [ a mod n ] with gcd( a, n

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