Show that $L(HK) = L(H) + L(K) - L(H\cap K) $ where L is the length of the composition series of the group and H and K are normal subgroups of G.

Question

I already proved that $L(G) = L(H) + L(\dfrac{G}{H})$ and then Im asked to prove that $L(HK) = L(H) + L(K) - L(H\cap K) $ but I fail to see a connection with what I already proved.

Answer 1

Hint: Remember that $$\frac{HK}{K}\simeq\frac{H}{H\cap K}$$ and shows that $L(M/N)=L(M)-L(N)$ for any subgroup $N$ of the group $M$.