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Unformatted text preview: H ⊆ G be a subgroup of order 24. Suppose that there is at least one left coset of H in G (other than H itself) that is also a right coset of H in G . Prove that H is a normal subgroup of G . 4 4. Let G be a group and let H be a subgroup of G such that the index ( G : H ) is ﬁnite. Prove that there is a normal subgroup H of G such that H ⊆ H and such that ( G : H ) is ﬁnite. Show further that there is an n ≥ 1 so that g n ∈ H for all g ∈ G . 5 5. Let G be a ﬁnite group, and let H be a normal subgroup of G . Let P be a pSylow subgroup of H , and let K be the normalizer of P in G . Establish the equality G = HK . 6...
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 Spring '10
 tao/analysis
 Math, Normal subgroup, KENNETH A. RIBET

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