4023f10ps5

# 4023f10ps5 - Homework #5 Due: October 13, 2010 1. Decide...

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Homework #5 Due: October 13, 2010 1. Decide whether each of the following sets forms a group with respect to the given operation. If it does form a group, give the identity element and the inverse of each element. If it does not form a group, identify which of the group axioms fail. (a) Q / = n m n : m; n 2 Z ; m 6 = 0 ; n 6 = 0 o with the operation of multiplication. (b) 1 ; 1 g ‰ Z with the operation of multiplication. (c) Z with the operation of subtraction. (d) G = ‰• 1 n 0 1 : n 2 Z ± with the operation of matrix multiplication. (e) The set F of all functions from f 1 ; 2 ; 3 g to itself with the operation of composition of functions. 2. (a) Find elements a , b , and c of the group S 3 where ab = bc , but a 6 = c . (b) Find elements a and b of the group S 3 such that ( ab ) ¡ 1 6 = a ¡ 1 b ¡ 1 . (c) Find elements a and b of the group S 3 such that ( ab ) 2 6 = a 2 b 2 . 3. Suppose that Q is the group deﬂned by the following multiplication table. That is, the multiplication xy is determined by ﬂnding the element which is in the row starting with x

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4023f10ps5 - Homework #5 Due: October 13, 2010 1. Decide...

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