College Algebra Exam Review 129

College Algebra Exam Review 129 - 2.7. QUOTIENT GROUPS AND...

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

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

Unformatted text preview: 2.7. QUOTIENT GROUPS AND HOMOMORPHISM THEOREMS 139 a normal subgroup N , there is a surjective homomorphism with N as ker- nel, and, on the other hand, a surjective homomorphism is essentially de- termined by its kernel. Theorem 2.7.6 also reveals the best way to understand a quotient group G=N . The best way is to find a natural model, namely some naturally defined group G together with a surjective homomorphism ' W G ! G with kernel N . Then, according to the theorem, G=N G . With this in mind, we take another look at the examples given above, as well as several more examples. Example 2.7.7. Let a be an element of order n in a group H . There is a homomorphism ' W Z ! H given by k 7! a k . This homomorphism has range h a i and kernel n Z . Therefore, by the homomorphism theorem, Z =n Z h a i . In particular, if D e 2 i=n , then '.k/ D k induces an isomorphism of Z =n Z onto the group C n of n th roots of unity in C ....
View Full Document

Ask a homework question - tutors are online