Give a one-sentence synopsis of the proof of Theorem 10 .10. 

Data from Theorem 10.10

Let H be a subgroup of a finite group G. Then the order of H is a divisor of the order of G. 

Proof

Let n be the order of G, and let H have order m. The preceding boxed statement shows that every coset of H also has m elements. Let r be the number of cells in the partition of G into left cosets of H. Then n = rm, so m is indeed a divisor of n.