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Math Help - quaternions and Diheral group

  1. #1
    ux0
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    quaternions and Diheral group

    Prove that the quaternions \bold Q and the Dihedral group D_8 are non-isomorphic groups of order 8.
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    An isomorphism preserves the order of an element. Therefore if, for example, there are strictly more elements of order 2 in D_4 than in \bold Q, then theese groups are not isomorphic.
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  3. #3
    ux0
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    ok well i know that there is only one element in \bold Q of order 2, which is -I, and i know it is normal, how do I figure out how many subgroups of order 2 are in D_8 ?
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    The dihedral group of order 8 can be seen as the group of symmetries of the square. There are two diagonal symetries, which are involutions, therefore D_8 (with your notation) contains at least 2 elements of order 2; that is sufficient to prove what you want.

    D_8 has exactly 5 elements of order 2 (two diagonal symmetries, 2 lateral symmetries and one central symmetry) and 2 elements of order 4 (one rotation of \frac{\pi}{2} and its inverse).
    Last edited by clic-clac; October 27th 2009 at 07:22 AM. Reason: involutions and not idempotent...
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  5. #5
    MHF Contributor Swlabr's Avatar
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    Quote Originally Posted by clic-clac View Post
    There are two diagonal symetries, which are idempotent...
    Maybe I am missing something here, but the only idempotent in a group is the identity - an idempotent is an element a such that a^2=a. In a group we can cancel this to get that a=1...
    Last edited by Swlabr; October 27th 2009 at 08:56 AM.
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  6. #6
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    You're not missing anything, I wrote idempotent thinking of involution... My bad
    Thanks for pointing out the error.
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