hw05 - Exercise 4 Let G be a group in which every element...

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Modern Algebra 1 Homework 3.1 Spring 2010 Due February 3 Exercise 1. What is the largest possible order of an element of S 7 ? Exercise 2. Let σ S n and let i ∈ { 1 , 2 ,...,n } . We say that i is a fixed point of σ if σ ( i ) = i . Prove that if σ is a cycle containing i and i is a fixed point of σ k , then the length of σ divides k . Exercise 3. Let G be a group and let a,b G satisfy ab = ba . If | a | = m and | b | = n show that | ab | divides the least common multiple of m and n . Give an example of this situation in which | ab | is not equal to the least common multiple.
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Unformatted text preview: Exercise 4. Let G be a group in which every element has order less than or equal to 2. Show that G is abelian. [ Hint: Consider ( ab ) 2 for arbitrary a,b ∈ G .] Exercise 5. The exponent of a group G is the smallest n ∈ N so that a n = e for all a ∈ G . a. Show that the exponent of G is the least common multiple of the orders of the elements in G . b. Find the exponents of S 3 ,S 4 and S 5 ....
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