ahw3 - Math 4124 Wednesday, February 8 Third Homework...

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Math 4124 Wednesday, February 8 Third Homework Solutions 1. 1.5.2 on page 36. Write out the group table for Q 8 . 1 - 1 i - i j - j k - k 1 1 - 1 i - i j - j k - k - 1 - 1 1 - i i - j j - k k i i - i - 1 1 k - k - j j - i - i i 1 - 1 - k k - i i j j - j - k k - 1 1 i - i - j - j j k - k 1 - 1 - i i k k - k j - j - i i - 1 1 - k - k k - j j i - i 1 - 1 2. 1.6.7 on page 40. Prove that D 8 and Q 8 are not isomorphic. The group D 8 has one element of order 1, two elements of order 4, and five elements of order 2. In fact if we write D 8 = { r , s | r 4 = s 2 = e , rs = sr - 1 } , so that the elements of D 8 are r i and sr i where i = 0 , 1 , 2 , 3, then r 0 has order 1, r and r 3 have order 4, and sr i (for i = 0 , 1 , 2 , 3) and r 2 have order 2. On the other hand Q 8 has one element of order 1, one element of order 2 (namely - 1), and six elements of order 4. If D 8 and Q 8 were isomorphic, then they would have the same number of elements of order 2 (and also the same number of elements of order 4); since this is not the case, we see that
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