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    Question 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.
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    Quote Originally Posted by r7iris View Post
    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.
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    Quote Originally Posted by r7iris View Post
    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:
    I = \left [ \begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix} \right ]

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

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

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

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

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

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

    \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 \{ I, \tau \}, where \tau is one of the reflections.

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

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

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

    You take a look at the other reflection subgroups.

    -Dan
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