## Review of Group Theory: The Third Isomorphism Theorem

**Point of post: **In this post we discuss the *Third Isomorphism Theorem*.

*Motivation*

The third isomorphism theorem has to deal with the situation when with . It asks us if the simple arithmetic fact somehow applies to groups. In essence, it asks if ?

*The Third Isomorphism Theorem*

We begin with a lemma:

**Lemma: ***Let with . Then, . *

**Proof: **Merely note that the canonical projection

is a surjective homomorphism, , and . The result follows from an earlier theorem.

Now, the rest is easy:

**Theorem (Third Isomorphism Theorem): ***Let be a group with and . Then,*

**Proof: **This is simple as noticing that

is evidently a surjective homomorphism and

Thus, by the First Isomorphism Theorem we may conclude that

as desired.

**References:**

1. Lang, Serge. *Undergraduate Algebra*. 3rd. ed. Springer, 2010. Print.

2. Dummit, David Steven., and Richard M. Foote. *Abstract Algebra*. Hoboken, NJ: Wiley, 200

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