Let H,K subgroups of G for each x in G define the double cost HK of x

a) Prove that HxK is the union of the left costs

where

is the orbit containing xK of H acting by left Multiplication on the set of left costs of K

My work

I want to prove that

it is clear that

so

let

hx in G so hxK in the union

d) Prove that

my question is about d) I put a) because I think it will help in solving d)

I was thinking about the order of orbit equal to the index group of the stabilizer

so we will have

which is not like what i want to prove

THanks