Adv Alegbra HW Solutions 51

Adv Alegbra HW Solutions 51 - 51 2.42 Let G = GL(2 Q and...

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51 2.42 Let G = GL ( 2 , Q ) , and let A = £ 0 1 10 ± and B = £ 01 11 ± . Show that A 4 = I = B 6 , but that ( AB ) n ±= I for all n > 0. Conclude that can have in f nite order even though both factors A and B have f nite order. Solution. = · ¸ and ( ) n = · 1 n ¸ . 2.43 (i) Prove, by induction on k 1, that · cos θ sin θ sin θ cos θ ¸ k = · cos k θ sin k θ sin k θ cos k θ ¸ . Solution. The proof is by induction on k . The base step is obvious. For the inductive step, let A = · cos θ sin θ sin θ cos θ ¸ . Then A k + 1 = AA k , and matrix multiplication gives the desired result if one uses the addition formulas for sine and cosine. (ii) Find all the elements of f nite order in SO ( 2 , R ) , the special or- thogonal group. Solution. By part (i), a matrix A = · cos θ sin θ sin θ cos θ ¸ . has f nite order if and only if cos k α = 1 and sin k α = 0; that is, when k α is an integral multiple of 2 π . Thus, A has f nite order if α = 2 π/ k for some nonzero integer k . 2.44 If G is a group in which x 2 = 1 for every x G , prove that G must be
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This note was uploaded on 12/21/2011 for the course MAS 4301 taught by Professor Keithcornell during the Fall '11 term at UNF.

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