2nd Isomorphism Theorem (Lattice Diagram)

I found this “lattice diagram”  only in an old Chinese Abstract Algebra Textbook, never seen before in any American/UK  or in French textbooks . Share here with the students who would find difficulty remembering the 3 useful Isomorphism Theorems.

Reference: 2nd Isomorphism Theorem (“Diamond Theorem”)

Let G be a group. Let H be a subgroup of G, and let N be a normal subgroup of G. Then:

1. The product HN is a subgroup of G,
The intersection H ∩ N is a normal subgroup of H, and

2. The 2 quotient groups
(HN) / N and
H / (H∩ N)
are isomorphic.

It is easy to remember using the green diagram below: (similarly can be drawn for 1st & 3rd Isomorphism)

This 2nd isomorphism theorem has been called the “diamond theorem” due to the shape of the resulting subgroup lattice with HN at the top, H∩ N…