Second Isomorphism Theorem: Statement, Proof

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.

Second Isomorphism Theorem Statement

Let G be a group such that

1. H be a subgroup of G
2. 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(h₁h₂) = h₁h₂K = (h₁K) (h₂K) = φ(h₁) φ(h₂), 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 Group Orbit 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

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