math 435 homework 1

math 435 homework 1 - Homework 1 for MATH 435 Solutions to...

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Unformatted text preview: Homework 1 for MATH 435 Solutions to selected exercises Problem 1 Book, p. 146, Exercise 2.36 ( The answers are in the back of the back, but give the reasons. ) Solution. (i) FALSE, e.g. ( 2 2 ) 3 = 2 6 = 64 is not the same as 2 ( 2 3 ) = 2 8 = 256. (ii) FALSE, e.g. S 3 , the symmetric group of { 1, 2, 3 } . (iii) TRUE. Multiplication of to positive real numbers yields a positive real number, and the operation is clearly associative. The identity is 1, the inverse of a is 1 /a , which is positive i ff a is. (iv) FALSE, e.g. no positive real number has a positive additive inverse. (v) FALSE. This is true i ff G is abelian, since aba- 1 b- 1 = 1 aba- 1 = b ab = ba . But not every group is abelian. Problem 2 Let G be a group. Assume that for any a G , aa = e . Show that G is abelian. Solution. We have e = ( ab )( ab ) , hence a = aabab , hence ab = aababb , and since aa = e and bb = e , we have ab = ba ....
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math 435 homework 1 - Homework 1 for MATH 435 Solutions to...

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