Math505HWsolutions - MATH 405/505 ABSTARCT ALGEBRA HOMEWORK...

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: MATH 405/505 ABSTARCT ALGEBRA HOMEWORK MAIN IDEAS OF SOLUTIONS 2.65 Prove that U ( I 9 ) is isomrophic to I 6 and that U ( I 15 ) is isomorphic to I 4 I 2 . Solution: For the first part note U ( I 9 ) contains 6 elements. Now exhibit an element of order 6. For the second part note that U ( I 15 ) contains 8 elements. Now exhibit an element a of order 4 and an element b of order 2 so that h a i h b i = 1 and h a ih b i = U ( I 15 ). 2.67 Prove that (i) Aut ( V ) = S 3 , (ii) Aut ( S 3 ) = S 3 , and (iii) Aut ( Z ) = I 2 . Solution: (i) Note that V = I 2 I 2 , so V = { 0 = (0 , 0) ,a = (1 , 0) ,b = (0 , 1) ,c = (1 , 1) } . Every automorphism of V fixes 0 and permutes a , b , c . This shows that Aut ( V ) is a subgroup of S X , where X = { a,b,c } . Now check that every permutation of X does also define an automorphism. (ii) Note that S 3 contains 3 elements of order 2, x = (12), y = (13), and z = (23). An element of Aut ( S 3 ) permutes these 3 elements, so Aut ( S 3 ) is a subgroup of S X , where X = { x,y,z } . Now check that every permutation of X does also define an automorphism. (iii) Every automophism of Z send a generator to a generator. The result follows from the fact that the only generators of Z are 1. 2.68 If G is a finite group such that Aut ( G ) is the trivial group, then G is trivial or G = I 2 . Solution: Let g be the inner automorphism that is conjugation by g , so g ( g ) = gg g- 1 for all g G . Since Aut ( G ) is trivial, i.e. only containes the identity, g is the identity.is the identity....
View Full Document

This note was uploaded on 01/18/2012 for the course INFORMATIK 2011 taught by Professor Phanthuongcang during the Winter '11 term at Cornell University (Engineering School).

Page1 / 3

Math505HWsolutions - MATH 405/505 ABSTARCT ALGEBRA HOMEWORK...

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