Review of Group Theory: The Third Isomorphism Theorem
Point of post: In this post we discuss the Third Isomorphism Theorem.
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
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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