I am reading a proof of the Jordan Holder theorem given here. (relevant part is given below)
When the author says that $K_i= K\cap G_i$ in the second paragraph, I do not understand the proof of how $K_i/K_{i+1}$ is a normal subgroup of $G_i/G_{i+1}.$ First I am not sure which isomorphism theorem is being used here. Furthermore, I am not sure how one can say the image is normal and the kernel is $K_{i+1}.$ Perhaps someone can offer an explanation.
