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    What are the symmetries of a cube and b) a regular cylinder?
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    Sorry can you expand on what you mean in the question, or say what level you are at (school or uni). I'm just unsure whether you mean planes of symmetry, rotational symmetry rotations or the members of the symmetry group for the shapes mentioned.
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    I think it wants the members of the symmetry group for the shapes. thank you
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    (Original post by Goldenratio)
    I think it wants the members of the symmetry group for the shapes. thank you
    In that case for a cube you can choose one corner and rotate the cube so that the chosen corner is in any of the 8 possible positions, you can then choose a adjacent corner and move that to any of the 3 adjacent corner positions by rotation, then with these two fixed you can then have a reflection in the plane going diagonally through the cube, the line being between the two specified vertices being in the plane.

    This gives you 8x3x2=48 possibilities.

    You obviously have infinite for a cylinder, these consist of any sized rotation about an axis through the centre of the cross section (infinite number), then either followed or not by a reflection.
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    Let V be the set of vertices of the cube. The stabiliser of any element in V contains 3 rotations and 3 reflections. And its orbit is the whole of V (of size 8). So order of the symmetry group is 6x8=48. The rotation group has order 3x8=24 and the elements permute the four diagonals of the cube so this is isomorphic to S4. I think the whole symmetry group is isomorphic to S4xZ2.
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    Wow fantastic! It's lovely to be able to see one method and then another fully explained! Thank you so much!
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    I bet you a jam tart that the symmetry group of the cylinder is SL(2,R)xZ2
 
 
 
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