Abstract Nonsense

Crushing one theorem at a time

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 D_{n} be the Dihedral Group with presentation

\text{ }

\displaystyle D_{n}=\left\langle r,s:r^n=e,\;\;s^2=e,\;\;srs=r^{-1}\right\rangle

\text{ }

then D_{n}\cong \mathbb{Z}_n\rtimes_\varphi\mathbb{Z}_2 where \varphi(1) is the inversion map i(z)=z^{-1} (which is an automorphism since \mathbb{Z}_n is abelian).

Proof: Consider \left\langle r\right\rangle\overset{\text{def.}}{=}R. It’s clear that R\unlhd D_{n} since \left(D_{n}:R\right)=2 (appealing to an earlier theorem). But, it’s also clear that if \left\langle s\right\rangle\overset{\text{def.}}{=}F then F\cap R=\{e\}. Thus, since by definition \text{D}_{n}=RS we may conclude by our previous theorem that

\text{ }

D_{n}\cong R\rtimes_\varphi F

 \text{ }

where

\text{ }

\psi:F\longrightarrow R:f\mapsto i_f

\text{ }

where i_f(r)=frf^{-1}. Note though that by definition for all r\in R

\text{ }

i_e(r)=r

 \text{ }

and

\text{ }

i_s(r)=srs^{-1}=r^{-1}

 \text{ }

So, define

\text{ }

\phi:R\rtimes_\psi F\longrightarrow \mathbb{Z}_n\rtimes_\varphi\mathbb{Z}_2:\left(r^i,s^j\right)\mapsto (i,j)

\text{ }

Evidently \phi is a bijection and note that

\text{ }

\phi\left((r^i,s^j)(r^{m},s^k)\right) = \phi\left(r^i\psi_{s^j}\left(r^m\right),s^{j+k}\right)

\text{ }

but

\text{ }

\displaystyle \left(r^i\psi_{s^j}\left(r^m\right),s^{j+k}\right)=\begin{cases}\left(i+m,k\right) & \mbox{if} \quad j=0 \\ (i-m,1+k) & \mbox{if} \quad j=1\end{cases}

 \text{ }

note though that

\text{ }

\phi\left((i,j)\right)\phi\left((m,k)\right)=(i,j)(m,k)=\left(i+\psi_j(m),j+k\right)

\text{ }

but

\text{ }

\displaystyle \left(i+psi_j(m),j+k\right)=\begin{cases}\left(i+m,k\right) & \mbox{if}\quad j=0\\ \left(i-m,1+k\right) & \mbox{if}\quad j=1\end{cases}

\text{ }

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

\text{ }

D_{n}\cong R\rtimes_\psi F\cong \mathbb{Z}_n\rtimes_\varphi\mathbb{Z}_2

 \text{ }

and so the conclusion follows. \blacksquare

\text{ }

\text{ }

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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January 11, 2011 - Posted by | Algebra, Group Theory | , ,

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