ma5c-hw8-soln

ma5c-hw8-soln - Ma 5 c HOMEWORK 8 SOLUTION SPRING 09 The...

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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 G-stable. Therefore W is a FG-submodule 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 ....
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This note was uploaded on 07/19/2010 for the course MA 5c taught by Professor Susamaagarwala during the Spring '09 term at Caltech.

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ma5c-hw8-soln - Ma 5 c HOMEWORK 8 SOLUTION SPRING 09 The...

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