# hw02 - is called a semigroup Prove that G is a group if and...

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Modern Algebra 1 Homework 1.2 Spring 2010 Due January 20 Exercise 4. Let G be a group. Use induction to prove that if a 1 ,a 2 ,...,a n G then ( a 1 a 2 ··· a n ) - 1 = a - 1 n ··· a - 1 2 a - 1 1 . Exercise 5. Prove that | S n | = n ! and that S n is non-abelian if n 3. Exercise 6. Find an element σ S 5 so that σ 6 = ± (the identity), but σ i 6 = ± for i = 1 , 2 , 3 , 4 , 5. Exercise 7. Let G be a nonempty set with an associative binary operation (such an object
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Unformatted text preview: is called a semigroup ). Prove that G is a group if and only if for every a,b ∈ G the equations ax = b and ya = b have unique solutions x,y ∈ G . [ Hint: Use the “one-sided” theorem proven in class.]...
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## This note was uploaded on 02/29/2012 for the course MATH 3362 taught by Professor Ryandaileda during the Spring '10 term at Trinity University.

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