18 chapter 4 abelian groups this chapter reviews the

Info icon This preview shows pages 23–26. Sign up to view the full content.

View Full Document Right Arrow Icon
18
Image of page 23

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
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
Image of page 24
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
Image of page 25

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 26
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern