Adv Alegbra HW Solutions 49

Adv Alegbra HW Solutions 49 - 49 2.36 True or false with...

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Unformatted text preview: 49 2.36 True or false with reasons. (i) The function e : N × N → N, defined by e(m , n ) = m n , is an associative operation. Solution. False. (ii) Every group is abelian. Solution. False. (iii) The set of all positive real numbers is a group under multiplication. Solution. True. (iv) The set of all positive real numbers is a group under addition. Solution. False. (v) For all a , b ∈ G , where G is a group, aba −1 b−1 = 1. Solution. False. (vi) Every permutation of the vertices v1 , v2 , v3 of an equilateral triangle π3 is the restriction of a symmetry of π3 . Solution. True. (vii) Every permutation of the vertices v1 , v2 , v3 , v4 of a square π4 is the restriction of a symmetry of π4 . Solution. False. (viii) If a , b ∈ G , where G is a group, then (ab)n = a n bn for all n ∈ N. Solution. False. (ix) Every infinite group contains an element of infinite order. Solution. False. (x) Complex conjugation permutes the roots of every polynomial having real coefficients. Solution. True. 2.37 If a1 , a2 . . . , an are (not necessarily distinct) elements in a group G , prove that −− − (a1 a2 · · · an )−1 = an 1 · · · a2 1 a1 1 . Solution. Absent. 2.38 (i) Compute the order, inverse, and parity of α = (1 2)(4 3)(1 3 5 4 2)(1 5)(1 3)(2 3). Solution. α = (1 2)(4 3)(1 3 5 4 2)(1 5)(1 3)(2 3) = (1 5 4)(2 3). ...
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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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