Order of the Elements in $D_4$ 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? 
 A: 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.
A: 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 $.
