Order of the Elements in $D_4$

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I am having trouble finding the order of each element in $D_4$.

I know $D_4 = \{R_0, R_{90}, R_{180}, R_{270}, V, H, D, D'\}$ and $|D_4| = 8$.

How would I go about finding the order of the elements?

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2 Answers

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Start with the definition of order, which loosely means in this context: "how many times can you iterate one of the operations before everything is back to how it started?" For instance, $R_{90}$ refers to a rotation by 90 degrees. If we do that 4 times, our object is back to how it started, so the order of that element is 4.

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The Dihedral Group D4 has representation $ \{I,A,B,C,D,E,F,G\} $ which
$ I= \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} $,
$ A= \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix} $,
$ B= \begin{bmatrix} -1 & 0 \\ 0 & -1 \end{bmatrix} $,
$ C= \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix} $,
$ D= \begin{bmatrix} -1 & 0 \\ 0 & 1 \end{bmatrix} $,
$ E= \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} $,
$ F= \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix} $,
$ G= \begin{bmatrix} 0 & -1 \\ -1 & 0 \end{bmatrix} $

By multiplication of Matrices, the order of each element is
$ |I|=1 $, $ |A|=4 $, $ |B|=2 $, $ |C|=4 $, $ |D|=2 $, $ |E|=2 $, $ |F|=4 $, $ |G|=2 $.

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