{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

College Algebra Exam Review 125

# College Algebra Exam Review 125 - 2.7 QUOTIENT GROUPS AND...

This preview shows page 1. Sign up to view the full content.

2.7. QUOTIENT GROUPS AND HOMOMORPHISM THEOREMS 135 Example 2.7.2. (Finite cyclic groups as quotients of Z ). The construction of Z n in Section 1.7 is an example of the quotient group construction. The (normal) subgroup in the construction is n Z D f `n W ` 2 Z g . The cosets of n Z in Z are of the form k C n Z D OEkŁ ; the distinct cosets are OE0Ł D n Z ; OE1Ł D 1 C n Z ; : : : ; OEn D n 1 C n Z . The product (sum) of two cosets is OEaŁ C OEbŁ D OEa C . So the group we called Z n is precisely Z =n Z . The quotient homomorphism Z ! Z n is given by k 7! OEkŁ . Example 2.7.3. Now consider a cyclic group G of order n with generator a . There is a homomorphism ' W Z ! G of Z onto G defined by '.k/ D a k . The kernel of this homomorphism is precisely all multiples of n , the order of a ; ker .'/ D n Z . I claim that ' “induces” an isomorphism Q ' W Z n ! G , defined by Q '.OEkŁ/ D a k D '.k/ . It is necessary to check that this makes sense (i.e., that Q ' is well defined) because we have attempted to define the value of Q ' on a coset OEkŁ
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• 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.

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

• 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.

Dana University of Pennsylvania ‘17, Course Hero Intern

• 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.

Jill Tulane University ‘16, Course Hero Intern