On this page, we will learn about the third isomorphism theorem for groups along with its statement and proof.

Table of Contents

## Third Isomorphism Theorem Statement

Let G be a group and H, K be its two normal subgroups such that H ≤ K. Then we have a group isomorphism

(G/H)/(K/H) ≅ G/K.

## Third Isomorphism Theorem Proof

**First**, we prove that K/H is a normal subgroup of G/H. For gH∈ G/H and kH ∈ K/H, we have that

gH kH (gH)^{-1}

= gH kH g^{-1} H

= gkg^{-1} H

∈ K/H as gkg^{-1}∈K. This is because K is normal in G.

This shows that K/H is normal in G/H.

**Next**, we prove the group isomorphism. Let us define a mapping

φ: G/H → G/K

by φ(gH) = gK for g ∈ G.

Let us show that φ is well-defined. For g_{1}, g_{2}∈G with g_{1}H=g_{2}H, we have that

g_{1} = g_{2}h for some element h∈H.

As H ≤ K, we have h∈K. Thus, we can conclude that

g_{1}K=g_{2}K

⇒ φ(g_{1}H) = φ(g_{2}H).

This shows that φ is well-defined.

By definition, φ is onto.

Now, Ker φ = {gH∈ G/H: φ(gH) = K}

= {gH∈ G/H: gK = K}

= {gH∈ G/H: g ∈ K}

= G/K.

Hence, by the first isomorphism theorem of groups, we obtain that

$\dfrac{G/H}{K/H} \cong G/K$.

This completes the proof of the third isomorphism theorem for groups.

**For Other Isomorphism Theorems Click Below:**

**Also Read:**

Group Theory: The basics of group theory are discussed here.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 HomomorphismGroup HomomorphismOrder of a Permutation: How to Find |

## FAQs

**Q1: What is the 3rd isomorphism theorem?**

Answer: Let G be a group. Let H, K be two normal subgroups of G. If H ≤ K, then we have a group isomorphism (G/H)/(K/H) ≅ G/K.