jan24 - Math 4124 Monday, January 24 January 24, Ungraded...

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Unformatted text preview: Math 4124 Monday, January 24 January 24, Ungraded Homework Exercise 1.2.1 on page 27 Compute the orders of each of the elements in the following groups. (a) D 6 (b) D 8 (c) D 10 We shall only give details for D 10 . (a) 1 has order 1; r , r 2 have order 3; and s , sr , sr 2 have order 2. (b) 1 has order 1; r , r 3 have order 4; r 2 and s , sr , sr 2 , sr 3 have order 2. (c) 1 has order 1; r , r 2 , r 3 , r 4 have order 5; and s , sr , sr 2 , sr 3 , sr 4 have order 2. Recall that every element of D 10 can be written uniquely in the form s i r j where i = , 1 and j = , 1 , 2 , 3 , 4. The order of 1 being 1 is obvious. Also ( r i ) 5 = r 5 i = ( r 5 ) i = 1 i = 1, so the order of r i divides 5. Since 5 is a prime number, it follows that if i = 1 , 2 , 3 , 4, then the order of r i is 5. Next sr i 6 = 1 ( i = , 1 , 2 , 3 , 4), so if we can prove that ( sr i ) 2 = 1, then it would follow that | sr i | = 2 and we would be finished. However by using rs = sr- 1 i-times, ( sr i ) 2 = s ( r i s ) r i = ssr- i r i = s 2 = 1 as required. Exercise 1.2.2 on page 27 Use generators and relations to show that if x is any element of D 2 n which is not a power of r , then rx = xr- 1 ....
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This note was uploaded on 01/02/2012 for the course MATH 4124 taught by Professor Staff during the Spring '08 term at Virginia Tech.

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jan24 - Math 4124 Monday, January 24 January 24, Ungraded...

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