Matrizen/2x2/Dritteldrehung und Spiegelungen/Gruppe/Aufgabe

a) Zeige, dass die drei reellen Matrizen

{\begin{pmatrix}1&0\\0&1\end{pmatrix}},\,{\begin{pmatrix}-{\frac {1}{2}}&-{\frac {\sqrt {3}}{2}}\\{\frac {\sqrt {3}}{2}}&-{\frac {1}{2}}\end{pmatrix}},\,{\begin{pmatrix}-{\frac {1}{2}}&{\frac {\sqrt {3}}{2}}\\-{\frac {\sqrt {3}}{2}}&-{\frac {1}{2}}\end{pmatrix}}

bezüglich der Matrizenmultiplikation eine Gruppe bilden.

b) Zeige, dass die sechs reellen Matrizen

{\begin{pmatrix}1&0\\0&1\end{pmatrix}},\,{\begin{pmatrix}-{\frac {1}{2}}&-{\frac {\sqrt {3}}{2}}\\{\frac {\sqrt {3}}{2}}&-{\frac {1}{2}}\end{pmatrix}},\,{\begin{pmatrix}-{\frac {1}{2}}&{\frac {\sqrt {3}}{2}}\\-{\frac {\sqrt {3}}{2}}&-{\frac {1}{2}}\end{pmatrix}},\,{\begin{pmatrix}1&0\\0&-1\end{pmatrix}},\,{\begin{pmatrix}-{\frac {1}{2}}&-{\frac {\sqrt {3}}{2}}\\-{\frac {\sqrt {3}}{2}}&{\frac {1}{2}}\end{pmatrix}},\,{\begin{pmatrix}-{\frac {1}{2}}&{\frac {\sqrt {3}}{2}}\\{\frac {\sqrt {3}}{2}}&{\frac {1}{2}}\end{pmatrix}}

bilden.

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