Let H and K be subgroups of a finite group G Prove that HK

Let H and K be subgroups of a finite group G. Prove that |HK| = |H||K|/|H K|.

Solution

We notice that |H||K| is the size of H × K.

Defining a map f from H × K to HK by f : (h, k) 7 hk.

We shall show that for each x HK, the number of preimages of x in H × K is |H K|; this clearly implies the result.

Suppose that f(h, k) = hk = x. Then for any g H K we have x =hgg^-1k = f(hg, g^-1k).

It follows that for every element hg of h(H K) there is a preimage of x in H × K whose H coordinate is hg, and so there are at least |H K| preimages.

Conversely, if f(h₁, k₁) is a preimage of x then hk = h₁k₁, and so h^-1h₁= kk₁^-1.

Setting g = h^-1h₁, it is clear that g H K, and that h₁ = hg. So all preimages of x have their H coordinate in h(H K), and so there are exactly |H K| of them

Hence

|HK|=|H||K|/|HK|