Solutions to (Selected) HW Problems
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چکیده
6. [This is simply an adapted proof of Theorem 2.4.2.] First observe that, the map φ(g) [for a fixed g ∈ G] is well defined: Hx = Hy iff xy−1 ∈ H iff xg−1gy−1 ∈ H iff (xg−1)(yg(−1)−1 ∈ H iff Hxg−1 = Hyg−1 iff φ(g)(Hx) = φ(g)(Hy). Note also, that since we have equivalencies in all steps above, we also have proved that φ(g) is one-to-one. Also, for a fixed g ∈ G, the map φ(g) : Hx 7→ Hxg is indeed a permutation of R(H). Since we have already proved that it is one-to-one, suffices to show it is onto: indeed, if Hx ∈ R(H), then φ(g)(H(xg)) = Hx. Let g, h ∈ G. To show that φ is a homomorphism, we need to prove φ(gh) = φ(g)◦φ(g) [as permutations of R(H)]. So, we need to show that for each Hx ∈ R(H), we have φ(gh)(Hx) = φ(g) ◦ φ(h)(Hx). But:
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Solutions to (Selected) HW Problems
Please read: I will try to post here a few solutions (or answers). The new solutions will be added to this same file. They might come with no explanation, just the “answer”. If yours do not match mine, you can try to figure out again. (Also, read the disclaimer below!) You can come to office hours if you want explanations for the unexplained answers. Be careful that just because our “answers” w...
متن کاملSolutions to (Selected) HW Problems
Please read: I will try to post here a few solutions (or answers). The new solutions will be added to this same file. They might come with no explanation, just the “answer”. If yours do not match mine, you can try to figure out again. (Also, read the disclaimer below!) You can come to office hours if you want explanations for the unexplained answers. Be careful that just because our “answers” w...
متن کاملSolutions to (Selected) HW Problems
Please read: I will try to post here a few solutions. The new solutions will be added to this same file. They might come with no explanation, just the “answer”. If yours do not match mine, you can try to figure out again. (Also, read the disclaimer below!) You can come to office hours if you want explanations for the unexplained answers. Be careful that just because our “answers” were the same,...
متن کاملSolutions to (Selected) HW Problems
Please read: I will try to post here a few solutions. The new solutions will be added to this same file. They might come with no explanation, just the “answer”. If yours do not match mine, you can try to figure out again. (Also, read the disclaimer below!) You can come to office hours if you want explanations for the unexplained answers. Be careful that just because our “answers” were the same,...
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