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# hw12 - Q by an element of Q yields a permutation in S 8 and...

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Modern Algebra 1 Homework 6.1 Spring 2010 Due February 24 Exercise 1. Let G be a group, let a G and let T a : G G be defined by T a ( x ) = ax . Prove that T a is a bijection. [ Suggestion: Find T - 1 a .] Exercise 2. Let Q = h A, B i = I, ± A, ± B, ± AB } (c.f. Homework 4.1, Exercise 1). Identify Q with { 1 , 2 , . . . , 8 } as follows: I A - I - A B AB - B - AB l l l l l l l l 1 2 3 4 5 6 7 8 Under this identification, left-multiplication in
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Unformatted text preview: Q by an element of Q yields a permutation in S 8 , and according to (the proof of) Cayley’s theorem the resulting collection of permutations is a subgroup of S 8 isomorphic to Q . Find all of the elements of this subgroup....
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