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Math Help - Factor groups for the quaternion group?

  1. #1
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    Factor groups for the quaternion group?

    I know the subgoups of the quaternion group (and that they are all normal), but i'm not sure how to determine the factor group Q8/N for each of the normal subgroups N.

    Thanks for your help.
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  2. #2
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    Quote Originally Posted by Louise View Post
    I know the subgoups of the quaternion group (and that they are all normal), but i'm not sure how to determine the factor group Q8/N for each of the normal subgroups N.

    Thanks for your help.
    if |N| = 1, 4 or 8, then Q_8/N \cong Q_8, \ C_2, or \{1 \} respectively. the only (normal) subgroup of order 2 is N=\{1,-1\}. then Q_8/N = \{\bar{1}, \bar{i}, \bar{j}, \bar{k} \} \cong C_2 \times C_2. (here C_2 is the cyclic group of order 2.)

    you probably know that V=C_2 \times C_2 is also called the Klein 4-group.
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  3. #3
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    How do you do the calculation Q8/{1,-1} to give {1,i,j,k}?
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  4. #4
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    Quote Originally Posted by Louise View Post
    How do you do the calculation Q8/{1,-1} to give {1,i,j,k}?
    Q_8/{1,-1} is NOT {1,i,j,k} but the images of these elements in the factor group, just Noncommalg denoted by putting a bar over the elements i,j,k.

    Why is this thing isomorphic with Z_2 x Z_2? Well, check that any of the elements 1, i, j, k squared gives you an element in {1,-1}...

    Tonio
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