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 f1 ( K ) = { x ∈ G : f ( x ) ∈ K } . Show that f1 ( K ) is a subgroup of G. Note: We are not assuming that f is a bijection, and therefore f1 does not mean “inverse function” here. Instead, f1 ( 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 mnm1 n1 . 1...
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 Spring '10
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 Math, Algebra, σ, τ, right cosets, left cosets

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