In this article, we will learn about the second isomorphism theorem for groups. This is also known as the **diamond isomorphism theorem**. The statement with its proof is provided below.

Table of Contents

## Second Isomorphism Theorem Statement

Let G be a group such that

- H be a subgroup of G
- K be a normal subgroup of G.

Then we have a group isomorphism

H/(H∩K) ≅ HK/K.

## Second Isomorphism Theorem Proof

The below steps have to be followed to prove the second isomorphism theorem for groups.

**Step 1:** To show HK is a subgroup of G.

In order to prove this, we need to show that HK=KH. For h ∈ H, k ∈ K, as K is normal in G, we have that hkh^{-1} ∈ K.

∴ hk = (hkh^{-1})h ∈ KH.

Hence, HK ⊆ KH.

In a similar way, KH ⊆ HK.

Therefore HK = KH, proving that HK is a subgroup of G.

**Step 2:** To show K is normal in HK.

As HK is a subgroup of G by Step 1, and K is normal in G by assumption, one can easily deduce that K is a normal subgroup of HK.

**Step 3:** To show H∩K $\trianglelefteq$ H. This follows from the fact that K $\trianglelefteq$ G.

We will now prove the isomorphism.

**Step 4:** Define a mapping

φ: H → HK/K

by φ(h) = hK for h ∈ H.

Note that φ(h_{1}h_{2}) = h_{1}h_{2}K = (h_{1}K) (h_{2}K) = φ(h_{1}) φ(h_{2}), so φ is a group homomorphism.

By the definition of φ, φ is onto.

Now, Ker φ = {h ∈ H: φ(h) = K}

= {h ∈ H: hK = K}

= {h ∈ H: h ∈ K}

= H∩K.

So by the first isomorphism theorem of groups, we can conclude that

H/(H∩K) ≅ HK/K.

This proves the second isomorphism theorem for groups.

**Also Read:**

Group Theory: The group theory is discussed here in detail.Order of a Group: The order of a group and its elements are discussed here with formulas. Abelian Group: The definition of an abelian group is discussed along with its properties and examples Center of a GroupOrbit Stabilizer Theorem: Its statement and proof are provided here.Cyclic Group: The definition, properties, and related theorems on cyclic groups are discussed.Kernel of a Group Homomorphism |

## FAQs on Second Isomorphism Theorem

**Q1: What is second isomorphism theorem for groups?**

Answer: Let G be a group and H, K be its two subgroups. If K is normal in G, then we have a group isomorphism H/(H∩K) ≅ HK/K.

**Q2: What is first isomorphism theorem for groups?**

Answer: Let G and H be two groups and Let φ: G → H be an onto homomorphism. Then we have a group isomorphism G/ker(φ) ≅ H.