Bourbaki

Exercise 6, § 5

Algebra, I, 141

Let H be a subgroup of finite index in a group G. Suppose that G is the union of the conjugates of H. Show that H = G. (Reduce it to the case where G is finite by means of Exercise 5. In the latter case, note that:

Card⁡(⋃x∈GxHx−1)≤1+(Card⁡(H)−1)Card⁡(G/H). \operatorname{Card}\left( \bigcup_{x \in G} x H x^{-1} \right) \leq 1 + (\operatorname{Card}(H) - 1)\operatorname{Card}(G/H).
Solution

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Transcribed from the printed volume. Not translated.