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solution11 - PMath 336 Introduction to Group Theory...

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PMath 336: Introduction to Group Theory Exercise set 11 July 25, 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 { 0 , . . . , 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 = 0 i i 0 and j = 0 1 - 1 0 . Questions 1. [10] Let π : G -→ H be a group homomorphism, and let s : H -→ G be a section. Prove that if for some h, h 0 H and g Ker ( π ), g - 1 s ( h ) g = s ( h 0 ), then h = h 0 . (This shows that in the criterion for products, theorem 145, we may replace the condition that s ( H ) commutes with the kernel, by the requirement that it is normal.) Solution: Applying π to both sides, we get h = π ( s ( h )) = π ( g - 1 ) π ( s ( h )) π ( g ) = π ( g - 1 s ( h ) g ) = π ( s ( h 0 )) = h 0 2. Let P be the group of matrices of the form x y 0 1 x , where x, y Q , and x 6 = 0. Let π : P -→ Q * be the function sending such a matrix to x . (a) [3] Prove that π is a group homomorphism (b) [5] Prove that the kernel of π is isomorphic to ( Q , +).

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