## Review of Group Theory: Semidirect Products (Pt. III The Dihedral Group)

**Point of post: **This is a continuation of this post.

We end our very short discussion of the very interesting topic of semidirect products with a corollary( more of an example than a corollary). To the interested reader one should consult a book like Grillet (see references) to go one step further and study *extension groups*. Alas, time doesn’t permit me to do this as of now.

**Corollary: ***Let be the Dihedral Group with presentation*

* *

*then where is the inversion map (which is an automorphism since is abelian).*

**Proof: **Consider . It’s clear that since (appealing to an earlier theorem). But, it’s also clear that if then . Thus, since by definition we may conclude by our previous theorem that

where

where . Note though that by definition for all

and

So, define

Evidently is a bijection and note that

but

note though that

but

from where it follows that is a homomorphism, and thus an isomorphism and so

and so the conclusion follows.

**References:**

1. Simon, Barry. *Representations of Finite and Compact Groups*. Providence, RI: American Mathematical Society, 1996. Print.

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

3.Grillet, Pierre A. *Abstract Algebra*. New York: Springer, 2007. Print.

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