Exercise 64. Let G be a group and let H be a subgroup of index n. If G| ≤n!, then G is not simple. Exercise 65. If H is a proper subgroup of An (n ≥ 5), then |G: H| ≥ n (Hint: use the fact that An is simple). Example. The group A5 has no proper subgroup of order greater than 12. In fact by Exercise 65, every subgroup of A5 has index at least 5. The smallest divisor of 60 is 12. Similarly, A6 of order 360 has no subgroup of order greater oro rothor small F

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