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Math Help - Subgroups

  1. #1
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    Subgroups

    1. Draw the lattice of subgroups (up to conjugacy) of S_{3} \times \mathbb{Z}_{2}.

    The symmetry of a system of balancing balls is given by
    S_{3} \times \mathbb{Z}_{2} are S_{3} \times \mathbb{Z}_{2} which can be thought of as the group \{id,(123),(321),(12),(13),(23),-id,-(123),-(321),-(12),-(13),-(23)\}

    Now I know that, up to conjugacy, the subgroups of S_{3} \times \mathbb{Z}_{2} are S_{3} \times \mathbb{Z}_{2}, \mathbb{Z}_{3} \times \mathbb{Z}_{2}=\{id,-id,(123),(321),-(123),-(321)\}, \mathbb{Z}_{2} \times \mathbb{Z}_{2}=\{id,-id,(12),-(12)\}, S_{3}=\{id,(123),(321),(12),(13),(23)\}, \mathbb{Z}_{3}=\{id,(123),(321)\}, \mathbb{Z}_{2}^{'}=\{id,(12)\}, \mathbb{Z}_{2}=\{id,-id\}, \widetilde{\mathbb{Z}_{2}}=\{id,-(12)\} and I={\id\}.

    However, I don't know how to construct a lattice of subgroups from this. How should it be structured?

    2. For each of the subgroups write down, and justify, its Fixed Point Subspace when acting on the two dimensional surface given by \{\mathbb{R}^{3}: x_{1}+x_{2}+x_{3}=0 \}.

    What is a Fixed Point Subspace? How do you find it?

    3. Draw the lattice of isotropy subgroups up to conjugacy.

    Same problem as in 1.

    If someone could enlighten me on this that would be great.
    Thanks!
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  2. #2
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    Quote Originally Posted by Primenumber View Post
    1. Draw the lattice of subgroups (up to conjugacy) of S_{3} \times \mathbb{Z}_{2}.

    The symmetry of a system of balancing balls is given by
    S_{3} \times \mathbb{Z}_{2} are S_{3} \times \mathbb{Z}_{2} which can be thought of as the group \{id,(123),(321),(12),(13),(23),-id,-(123),-(321),-(12),-(13),-(23)\}

    Now I know that, up to conjugacy, the subgroups of S_{3} \times \mathbb{Z}_{2} are S_{3} \times \mathbb{Z}_{2}, \mathbb{Z}_{3} \times \mathbb{Z}_{2}=\{id,-id,(123),(321),-(123),-(321)\}, \mathbb{Z}_{2} \times \mathbb{Z}_{2}=\{id,-id,(12),-(12)\}, S_{3}=\{id,(123),(321),(12),(13),(23)\}, \mathbb{Z}_{3}=\{id,(123),(321)\}, \mathbb{Z}_{2}^{'}=\{id,(12)\}, \mathbb{Z}_{2}=\{id,-id\}, \widetilde{\mathbb{Z}_{2}}=\{id,-(12)\} and I={\id\}.

    However, I don't know how to construct a lattice of subgroups from this. How should it be structured?
    The action of S_3 alone in the hyperplane seems to be straightforward. On the contrary, is it true that -(1 2)x_1=-x_2?

    I think you simply need to draw the subgroup lattice with a single bottom element \{id\} and the next level elements \{id, -id\}, \{id, (1 2)\}, and \{id, -(1 2)\}, and the top element S_3 \times \mathbb{Z}_2. You need to fill the intermediate elements between them.

    2. For each of the subgroups write down, and justify, its Fixed Point Subspace when acting on the two dimensional surface given by \{\mathbb{R}^{3}: x_{1}+x_{2}+x_{3}=0 \}.

    What is a Fixed Point Subspace? How do you find it?
    For instance, every subgroup of S_3 fixes \{\mathbb{R}^{3}: x_{1}+x_{2}+x_{3}=0 \}.

    3. Draw the lattice of isotropy subgroups up to conjugacy.

    Same problem as in 1.

    If someone could enlighten me on this that would be great.
    Thanks!
    This part may need some lengthy computations and take every combination of x_1, x_2, and x_3 into account, and their isotropy subgroups.
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