Understanding third isomorphism theorem with simple pictures.

Question

Recently I encountered the third isomorphism theorem in groups which says that if $G$ is a group and $H_1$ and $H_2$ be normal subgroups of $G$ such that $H_1 \subset H_2$,then show that $(G/H_1)/(H_2/H_1)\approx G/H_2$.But I cannot visualize it properly or understand its actual significance.I started dealing with group theory just a few days ago,so I am much more convenient with vector spaces rather than groups.So I was looking for a visual interpretation of the same theorem at first, so that I can better understand the theorem rather than just doing a textbook proof hard to understand.Can someone provide me some help?

Answer 1

See the diagram:

I have tried to give here a visual interpretation of what is actually going on here. First I demanded an answer from this site for this question but the motivation I got from the comments made me to think deeply and come up with my own intuition with which I am answering the question.First of all,$H$ and $K$ are normal subgroups of $G$ so we need not worry whether the quotients make sense or not,it can be taken for granted here that there is no problem with the quotient(in terms of well-definedness and forming a group structure).

Now Let us consider the group $G$ which is shown in the diagram as the largest rectangle.$H$ is depicted by the long rectangular strip at the leftmost part of $G$ and the twin strips adjoining $H$ are of course the cosets of it. Next,the subgroup $K$ is the small rectangle in $H$ shown by darker blue.Now $G/H$ which is shown as the oval in the left-down side of the diagram,where I have collapsed the cosets of $H$ in $G$ i.e. the members of $G/H$ to points.

Now come to the group $G/K$ ,to understand it first notice the cosets of $K$ in $G$ i.e. all the small rectangles looking 'twin' to $K$.Collapse each to a single point to get $G/K$.Now can you see that $H/K$ is a subset of $G/K$ because $H/K$ consists of cosets of $K$ in $H$ which are here the the two leftmost small rectangles (one is $K$ itself and the other is the small rectangle below $K$ and 'twin' to $K$ within $H$. So,in the diagram of $G/K$ notice that $H/K$ comprises the leftmost $2$ points in $G/K$ in the diagram.Quotienting essentially means collapsing to a single point(Notice by the way that cosets of $H/K$ in $G/K$ are all the point pair sets adjoining $H/K$ and twin to $H/K$).Now quotienting $G/K$ with $H/K$ will reduce each of these cosets (pair point sets) to a single point as shown in the right-down figure.Now don't the groups $G/H$ and $(G/K)/(H/K)$ look exactly identical,this accounts for the isomorphism.