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        riddles >> putnam exam (pure math) >> The D6 Group and Direct Products
        
(Message started by: Whiskey Tango Foxtrot on Nov 21st, 2006, 7:17pm)
Title: The D6 Group and Direct Products
Post by Whiskey Tango Foxtrot on Nov 21st, 2006, 7:17pm
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.
Title: Re: The D6 Group and Direct Products
Post by Barukh on Nov 24th, 2006, 5:34am
Let ap be a p-fold rotation about a certain axis a. For instance, x3 is a rotation about the x-axis through angle 2http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/pi.gif/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.
Title: Re: The D6 Group and Direct Products
Post by Whiskey Tango Foxtrot on Nov 26th, 2006, 8:24pm
Yes, that appears correct.


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