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Let G be a non-trivial finite group, and let H < G be a proper subgroup....

Let G be a non-trivial finite group, and let H < G be a proper subgroup. Let X be the set of conjugates of H, that is, X = {aHa^(−1) : a ∈ G}. Let G act on X by conjugation, i.e., g · (aHa^(−1) ) = (ga)H(ga)^(−1) .

Prove that this action of G on X is transitive.

Use the previous result to prove that G is not covered by the conjugates of H, i.e., G does not equal S a∈G aHa^(−1) .

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