Thread: group in square

1. group in square

Let G be the group D(4) of symmetries of a square and (tau) be any reflaction in G. Describe the left cosets of the subgroup {1, tau} of G.

2. Originally Posted by r7iris
Let G be the group D(4) of symmetries of a square and (tau) be any reflaction in G. Describe the left cosets of the subgroup {1, tau} of G.
You need to post your group table you are using. The way you have your question it can be anything.

3. Originally Posted by r7iris
Let G be the group D(4) of symmetries of a square and (tau) be any reflaction in G. Describe the left cosets of the subgroup {1, tau} of G.
The dihedral group (D4) has a 2D representation:
$\displaystyle I = \left [ \begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix} \right ]$

$\displaystyle R = \left [ \begin{matrix} 0 & -1 \\ 1 & 0 \end{matrix} \right ]$

$\displaystyle R^2 = \left [ \begin{matrix} -1 & 0 \\ 0 & -1 \end{matrix} \right ]$ <-- Sometimes -I or i. This is the inversion operator

$\displaystyle R^3 = \left [ \begin{matrix} 0 & 1 \\ -1 & 0\end{matrix} \right ]$ <-- Sometimes -R

$\displaystyle \tau_x = \left [ \begin{matrix} 1 & 0 \\ 0 & -1 \end{matrix} \right ]$ <-- In your notation. Usually $\displaystyle \sigma _x$

$\displaystyle \tau_y = \left [ \begin{matrix} -1 & 0 \\ 0 & 1 \end{matrix} \right ]$

$\displaystyle \tau _1 = \left [ \begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix} \right ]$ <-- Reflection over y = x

$\displaystyle \tau _2 = \left [ \begin{matrix} 0 & -1 \\ -1 & 0 \end{matrix} \right ]$ <-- Reflection over y = -x

We are interested in the left cosets of the subgroup $\displaystyle \{ I, \tau \}$, where $\displaystyle \tau$ is one of the reflections.

So, for example, consider the subgroup $\displaystyle \{ I, \tau _x \}$. The left cosets are:
$\displaystyle \{ g_i \} \cdot \{ I, \tau _x \}$
where the $\displaystyle \{ g_i \}$ represents the elements of D4.

So
$\displaystyle I \cdot \{ I, \tau _x \} = \{ I, \tau _x \}$
$\displaystyle R \cdot \{ I, \tau _x \} = \{ R, \tau _1 \}$
$\displaystyle R^2 \cdot \{ I, \tau _x \} = \{ R^2, \tau _y \}$
$\displaystyle R^3 \cdot \{ I, \tau _x \} = \{ R^3, \tau _2 \}$
$\displaystyle \tau _x \cdot \{ I, \tau _x \} = \{ \tau _x , I \}$
$\displaystyle \tau _y \cdot \{ I, \tau _x \} = \{ \tau _y, R^2 \}$
$\displaystyle \tau _1 \cdot \{ I, \tau _x \} = \{ \tau _1, R \}$
$\displaystyle \tau _2 \cdot \{ I, \tau _x \} = \{ \tau _2, R^3 \}$

So there are 4 distinct left cosets: $\displaystyle \{ I, \tau _x \}$, $\displaystyle \{ R, \tau _1 \}$, $\displaystyle \{ R^2, \tau _y \}$, and $\displaystyle \{ R^3, \tau _2 \}$.

You take a look at the other reflection subgroups.

-Dan

the left cosets of sub group of symmetries of asquare

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