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…**

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