Proposition 1.1. Let G be a Group and H and K be finite subgroups of G. Then |H||K| HK = |H NK In the next problem you p

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Proposition 1 1 Let G Be A Group And H And K Be Finite Subgroups Of G Then H K Hk H Nk In The Next Problem You P 1 (142.03 KiB) Viewed 16 times

Proposition 1.1. Let G be a Group and H and K be finite subgroups of G. Then |H||K| HK = |H NK In the next problem you prove this. Problem 1.4. Let G be a Group and H and K be finite subgroups of G. We have p: H x K → HK, p((h, k)) = hk + = - 1 == = . 1. Suppose that p((hı, kı)) = p((h2, ka)). Show that hīlh2 = kika? E H NK. 2. Let x € HK and choose hi e H, ki EK with x = hiki. Show that the map To : Tx HNK: +p-'(x), Tx(1) = (h11,1-1kı) is bijective. 3. Show that |H||K| = |HN K||HK| and hence = _ |HK| = |H||K| |H NK