Quiz2PracticeProblems

# - Quiz 2 Practice Problems Math 332 Spring 2010 Isomorphisms and Automorphisms 1 Let C be the group of complex numbers under the operation of

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Quiz 2 Practice Problems Math 332, Spring 2010 Isomorphisms and Automorphisms 1. Let C be the group of complex numbers under the operation of addition, and deﬁne a function ϕ : C C by ϕ ( a + bi ) = a - bi. Prove that ϕ is an automorphism of C . 2. Let G be an abelian group, and deﬁne a function ϕ : G G by ϕ ( a ) = a - 1 . Prove that ϕ is an automorphism of G . 3. Let ϕ : Z 10 U (11) be an isomorphism, and suppose that ϕ (1) = 8. Determine ϕ (3). 4. Prove that A 4 is not isomorphic to D 6 . 5. List the automorphisms of Z 8 . Express your answers as permutations of the set { 0 , 1 ,..., 7 } . 6. List the four elements of Inn( D 4 ). Express your answers as permutations of the set { e,r,r 2 ,r 3 ,s,rs,r 2 s,r 3 s } . 7. Let α Aut( D 5 ), and suppose that α ( r ) = r 2 and α ( s ) = rs . Find α ( r 3 s ). 8. Let α Aut( Q 8 ), and suppose that α ( i ) = k and α ( j ) = - i . Express α as a permutation of the set { 1 , - 1 ,i, - i,j, - j,k, - k } . 9. Determine the isomorphism type of the group whose Cayley table is shown below: e p q r s t u v e e p q r s t u v p p r u t q e v s q q s r v t u p e r r t v e u p s q s s v p u r q e t t t e s p v r q u u u q t s e v r p v v u e q p s t r 1

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Direct Products 10. Let G be a group, and deﬁne a function ϕ : G × G × G G × G × G by ϕ ( a,b,c ) = ( b,c,a ) . Prove that ϕ is an automorphism of G × G × G . 11.
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- Quiz 2 Practice Problems Math 332 Spring 2010 Isomorphisms and Automorphisms 1 Let C be the group of complex numbers under the operation of

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