Sol3 - Algebra 3 (2004-05) Solutions to Assignment 3...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

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

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

Unformatted text preview: Algebra 3 (2004-05) Solutions to Assignment 3 Instructor: Dr. Eyal Goren 1) If G,H are finite groups such that ( | G | , | H | ) = 1 then every group homomorphism f : G- H is trivial ( f ( G ) = { 1 } ). Proof. Let K be the kernel of f . Then | Im( f ) | = | G | / | K | divides both | H | and | G | . Hence | Im( f ) | = 1 and f is trivial. / 2) Find all possible homomorphisms Q- S 3 . Is there an injective homomorphism Q- S 4 ? (As usual, Q is the quaternion group of order 8). Solution: The image of such a homomorphism divides both | S 3 | = 6 and | Q | = 8. Thus, the image is either of cardinality 1 and then f is the trivial homomorphism, or of cardinality 2. In the latter case, the kernel of f is either h i i , h j i or h k i . If the homomorphism is not trivial then all the elements outside the kernel must go to the same element of order 2 in S 3 , which is a transposition. There are 3 such elements. Thus, we find that there are, besides the trivial homomorphism, 9 more homomorphisms; eachelements....
View Full Document

This note was uploaded on 11/19/2010 for the course ANTH 122 taught by Professor 323 during the Spring '10 term at Centennial College.

Page1 / 2

Sol3 - Algebra 3 (2004-05) Solutions to Assignment 3...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online