This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Ma 5 c HOMEWORK 8 SOLUTION SPRING 09 The exercises are taken from the text, Abstract Algebra (third edi tion) by Dummit and Foote, unless stated otherwise. Page 852, 3 . Let ρ : G → F be a degree 1 representation of G . Since imρ is a subgroup of an abelian group and therefore must be abelian, G/kerρ is abelian. Hence [ G,G ] ⊆ kerρ , i.e. every degree 1 represen tation of G corresponds to one of G/ [ G,G ]. The other direction follows from the canonical map G → G/ [ G,G ]. Page 852, 4 . Let W be the subspace of V generated by g i · v , then W is Gstable. Therefore W is a FGsubmodule of V . Since W is generated by  G  elements, it has dimension ≤  G  . Page 852, 6 . a) For S 3 , ϕ ( e ) = I 3 ϕ ((12)) = 0 1 0 1 0 0 0 0 1 , ϕ ((13)) = 0 1 1 0 1 0 1 0 0 , ϕ ((23)) = 1 0 0 0 0 1 0 1 0 , ϕ ((132)) = 0 1 0 0 0 1 1 0 0 , and ϕ ((123)) = 0 0 1 1 0 0 0 1 0 ....
View
Full
Document
This note was uploaded on 07/19/2010 for the course MA 5c taught by Professor Susamaagarwala during the Spring '09 term at Caltech.
 Spring '09
 SusamaAgarwala
 Algebra

Click to edit the document details