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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.
 Fall '11
 KeithCornell

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