# home2 - a σ A and a τ A Are the left cosets the same as...

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MATH 4107 HOMEWORK #2 DUE: February 12, 2010 Work the following problems and hand in your solutions. You may work together with other people in the class, but you must each write up your solutions independently. A subset of these will be selected for grading. Write LEGIBLY on the FRONT side of the page only, and STAPLE your pages together. 1. Let Z 8 = { 0 , 1 , 2 , 3 , 4 , 5 , 6 , 7 } under + mod 8 , and let G = { e, r, r 2 , r 3 , a, b, c, d } be the dihedral group of order 8 that we constructed in class (the set of symmetries of the square). Do there exist homomorphisms f : Z 8 Z and g : Z G such that g f is an isomorphism? (Remember, proof is required.) 2. In the group S 3 , define permutations σ and τ as follows (I show both the cycle and the two-row notation for these permutations): σ = (1 2 3) = parenleftbigg 1 2 3 2 3 1 parenrightbigg and τ = (1 2) = parenleftbigg 1 2 3 2 1 3 parenrightbigg
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Unformatted text preview: a σ A and a τ A . Are the left cosets the same as the right cosets? 3. Suppose that f : G → H is a homomorphism, and K is a subgroup of H. We de±ne the inverse image of K under f to be f-1 ( K ) = { x ∈ G : f ( x ) ∈ K } . Show that f-1 ( K ) is a subgroup of G. Note: We are not assuming that f is a bijection, and therefore f-1 does not mean “inverse function” here. Instead, f-1 ( K ) is shorthand for a particular subset of G. 4. Problem 2.4 #16. Hint: Consider the elements in G that are their own inverses, and the ones that are not. 5. Problem 2.5 #12. 6. Problem 2.5 #16. 7. Problem 2.5 #20. Hint: Think about the element mnm-1 n-1 . 1...
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