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Thread: Geometric symmetries

  1. #1
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    Exclamation Geometric symmetries

    This question tests your ability to describe symmetries geometrically and to represent them as permutations in cycle form. It also tests your understanding of conjugacy classes and their relationship to normal subgroups.

    The figure for this question is prism with three identical rectangular faces and an equilateral triangle at the top and base. The locations of the faces of the prism (numbered 1, 4 at the top and base and each side 2, 3 and 5, 5 being the back face) have been numbered so that we may represent the group G of all symmetries of the prism as permutations of the set {1,2,3,4,5}.

    a) Describe geometrically the symmetries of the prism represented in cycle form by (14)(23) and (25).

    b) Write down all the symmetries of the prism in cycle form as permutations of {1,2,3,4,5}, and describe each symmetry geometrically.

    c) Write down the conjugacy classes of G.

    d) Determine a subgroup of G of order 2, a subgroup of order 3, and a subgroup of order 4. In each case, state whether or not your choice of subgroup is normal, justifying your answer.
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  2. #2
    Super Member Rebesques's Avatar
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    Re: Geometric symmetries

    This question tests your ability to describe symmetries geometrically and to represent them as permutations in cycle form. It also tests your understanding of conjugacy classes and their relationship to normal subgroups.

    Cheating is one thing - Being good at cheating is another :P
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    Forum Admin topsquark's Avatar
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    Re: Geometric symmetries

    Quote Originally Posted by raggie29 View Post
    This question tests your ability to describe symmetries geometrically and to represent them as permutations in cycle form. It also tests your understanding of conjugacy classes and their relationship to normal subgroups.

    The figure for this question is prism with three identical rectangular faces and an equilateral triangle at the top and base. The locations of the faces of the prism (numbered 1, 4 at the top and base and each side 2, 3 and 5, 5 being the back face) have been numbered so that we may represent the group G of all symmetries of the prism as permutations of the set {1,2,3,4,5}.

    a) Describe geometrically the symmetries of the prism represented in cycle form by (14)(23) and (25).

    b) Write down all the symmetries of the prism in cycle form as permutations of {1,2,3,4,5}, and describe each symmetry geometrically.

    c) Write down the conjugacy classes of G.

    d) Determine a subgroup of G of order 2, a subgroup of order 3, and a subgroup of order 4. In each case, state whether or not your choice of subgroup is normal, justifying your answer.
    What have you been able to do so far?

    -Dan
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  4. #4
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    Re: Geometric symmetries

    You don't understand- the instructions for this test were "Find some one who can do these problems for you"!
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  5. #5
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    Re: Geometric symmetries

    Hey you guys...this problem was posted over 3 years ago;
    ya'll losing your time...
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  6. #6
    Super Member Rebesques's Avatar
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    Re: Geometric symmetries

    Quote Originally Posted by DenisB View Post
    Hey you guys...this problem was posted over 3 years ago;
    ya'll losing your time...

    ...Considering how we are all offering advice without charge on this board, I'd say you are spot on, D. :P


    But we could view this thread as a inner monologue between the elder teacher - who proclaims to have never cheated at anything - and his younger, lazy and sleazy, student self.
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