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Problem 8. Suppose that H has index 2 in G. Prove that H is normal in...

Problem 8. Suppose that H has index 2 in G. Prove that H is normal in G. (Hint: Usually to prove that a subgroup is normal, the conjugation criterion (Theorem 17.4) is easier to use than the definition, but this problem is a rare exception. Since H has index 2 in G, there are only two left cosets, one of which is H itself – use this to describe the other coset. Then do the same for right 1 cosets. Remembering that we need to prove that gH = Hg for every g ∈ G, consider two cases: g ∈ H and g /∈ H.)

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