Let G be a group and H be a subgroup of G Prove that if G H

Let G be a group and H be a subgroup of G. Prove that if G = H_a_1 H_a_2 ... H_a_n show that G = a_a^-1 H a_2^-1 H ... a_n^-1 H

Solution

The first equation is to be the right coset decomposition of G, (the problem statement does not define H_a explicitly).

Then the second equation is simply the left coset decomposition of G , taking into account

                 the set of inverses of Ha is the set aH (H is closed under inverses, as it is a subgroup of G)