113 HW 2_Math 113

# 113 HW 2_Math 113 - Mathematics 113 Homework 2 Curtis...

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Mathematics 113: Homework # 2 Curtis Tekell Dr. Poirier September 12, 2010 Exercise 1. Prove the following are or are not groups: (a) G = h Z , + i . (b) G = h Z , ×i . (c) G = h Z /n Z , ×i for some n Z + . (d) G = h ( Z /n Z ) × , ×i for some n Z + . (e) G = h M n × m ( R ) , + i . (f) G = h M n × m ( R ) , ×i . (g) G = h M n ( R ) , ×i . (h) G = h Q [ 2] , ×i . (i) G = h U, ×i , where U = { z C : n Z \ { 0 } 3 z n = 1 } . (j) G = h Q 0 , *i , where Q 0 is the set of rationals in lowest form, and a b * c d = a + c b + d , reduced to lowest terms. Proof. We assume associativity and commutativity of (standard) addition and multiplication. (a) Yes. 0 is the neutral element, and for n Z , n + ( - n ) = 0. (b) No. 0 G is not invertable. (c) No. If some m G is not relatively prime to n , it will not be invertable. (d) Yes. 0 is the neutral element, and for n Z , n + - n = 0 (e) Yes. The zero matrix O is the neutral element; for A G , A + ( - 1) A = O . (f) No. Not all

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113 HW 2_Math 113 - Mathematics 113 Homework 2 Curtis...

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