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# solution5 - PMath 336 Introduction to Group Theory Exercise...

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Unformatted text preview: PMath 336: Introduction to Group Theory Exercise set 5 June 20, 2008 Solution should be submitted by the end of the Monday lecture of the following week, either in class or in the submission box. You may not submit joint or identical works. Notation GL n ( S ) : Invertible n × n matrices with coefficients in S , under multiplication D n : The group of symmetries of the regular n-gon S n : The group of permutations of 1 ,...,n Z : The group of integers under addition Z n : The group { ,...,n- 1 } with addition modulo n U n : The group of numbers less than n and prime to n , with multiplication mod n . Q (quaternions): The subgroup of GL 2 ( C ) generated by i = i i and j = 0 1- 1 0 . Questions 1. Show that if G and H are isomorphic groups, then Aut ( G ) is isomorphic to Aut ( H ). Solution: Fix an isomorphism F : G-→ H . Given an automorphism f : G-→ G , we define a function T f : H-→ H by: T f ( h ) = F ( f ( F- 1 ( h ))). T f is an automorphism, since it is a composition of isomorphisms. Hence T : f 7→ T f defines a function from Aut ( G ) to Aut ( H ). We claim that T is a homomorphism. Indeed, T ( f ◦ g ) = F ◦ f ◦ g ◦ F- 1 = F ◦ f ◦ F- 1 ◦ F ◦ g ◦ F- 1 = T ( f ) ◦ T ( g ) Finally, T is invertible, since the function given by g 7→ F- 1 ◦ g ◦ F is the (two sided) inverse. 2. Recall that if A and B are sets, their Cartesian product A × B is the set of pairs ( a,b ) where a ∈ A and b ∈ B . If G and H are groups, we define an operation · on G × H by: ( a,b ) · ( c,d ) = ( ac,bd ) (note that the first product is in G and the second in H !) (a) Show that ( G × H, · ) is a group (b) Show that the projection map π 1 : G × H-→ G given by π 1 ( a,b ) = a is a surjective homomorphism,...
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solution5 - PMath 336 Introduction to Group Theory Exercise...

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