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Topic: The D6 Group and Direct Products (Read 607 times) |
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Whiskey Tango Foxtrot
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The D6 Group and Direct Products
« on: Nov 21st, 2006, 7:17pm » |
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Show that the group D6 generated by six-fold rotations about the z-axis and two-fold rotations about the x-axis is the direct product D3 x C2, where the principle axis of both D3 and C2 is the z-axis.
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Barukh
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Re: The D6 Group and Direct Products
« Reply #1 on: Nov 24th, 2006, 5:34am » |
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Let ap be a p-fold rotation about a certain axis a. For instance, x3 is a rotation about the x-axis through angle 2 /3 (counterclockwise).
Now, D6 – being a symmetry group of a regular hexagon – consists of transformations of the form z6nx2m, where n = 0, .., 5; m = 0, 1.
The direct product D3 x C2 (the latter being the rotation group of a digon) consists of ordered pairs (z3nx2m, z2k), where n = 0, … 2; m, k = 0, 1.
Both groups have order 12. We can easily establish the following bijection:
z61x20 <=> (z32x20, z21),
z61x21 <=> (z32x21, z21),
and all other elements are generated by these two.
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Whiskey Tango Foxtrot
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Re: The D6 Group and Direct Products
« Reply #2 on: Nov 26th, 2006, 8:24pm » |
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Yes, that appears correct.
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"I do not feel obliged to believe that the same God who has endowed us with sense, reason, and intellect has intended us to forgo their use." - Galileo Galilei
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