2.21:

Let H and K be subgroups of G. Suppose G is the disjoint union of subsets Ht_iK, for 1 <= i <= n. (These subsets are called double cosets.) We are to show |G:K| is the sum |H: H ^ t_i Kt_i^-1|, i ranging from 1 to n.

For simplicity set t = t_i, and write t_i Kt_i^-1 as K^t. The double coset HtK is the disjoint union of the left cosets htK of K, with h ranging over H. If h and h' are in H, the left cosets htK and h'tK are equal if and only if (ht)^-1h't is in K, which occurs if and only if h^-1h' is in H ^ K^t, by a straightforward calculation. This occurs if and only if h and h' lie in the same left coset of H ^ K^t in H. Then the number of left cosets htK in HtK is equal to the number of cosets of H ^ K^t in H, which is |H:H ^ K^t|. Apply the same calculation for each i, 1<=i<=n, and we have the desired equality.