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Math Help - D8 subgroups

  1. #1
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    D8 subgroups

    Easy question. How do I find the subgroups of D_8?

    D_8=<r,s:r^4=s^2=1, \ rs=sr^{-1}>

    D_8=\{1,r,r^2,r^3,s,sr,sr^2,sr^3\}
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  2. #2
    Super Member Bernhard's Avatar
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    Re: D8 subgroups

    See Deveno's posts in answer to my post "Normal Subgroups in D4 "

    Peter
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    Re: D8 subgroups

    Quote Originally Posted by Bernhard View Post
    See Deveno's posts in answer to my post "Normal Subgroups in D4 "

    Peter
    So the Centralizer of each element is a subgroup then?

    And normal subgroups are in the center?

    What about cyclic?
    Last edited by dwsmith; October 8th 2011 at 05:06 PM.
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  4. #4
    Super Member Bernhard's Avatar
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    Re: D8 subgroups

    Sorry.

    I was too quick in referrring you to my post - I was looking for normal subgroups only.

    Peter
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    Re: D8 subgroups

    Quote Originally Posted by Bernhard View Post
    Sorry.

    I was too quick in referrring you to my post - I was looking for normal subgroups only.

    Peter
    It didn't matter I learned something from it.

    So the cyclic subgroups are <r>=<r^3>, correct?
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  6. #6
    Super Member Bernhard's Avatar
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    Re: D8 subgroups

    Yes, but I think there may be others of order 2 such as <s> = {s, e} and <sr> = {sr, e}

    Do you agree?

    Peter
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  7. #7
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    Re: D8 subgroups

    Quote Originally Posted by Bernhard View Post
    Yes, but I think there may be others of order 2 such as <s> = {s, e} and <sr> = {sr, e}

    Do you agree?

    Peter
    How were you able to identify those at cyclic groups?
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  8. #8
    Super Member Bernhard's Avatar
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    Re: D8 subgroups

    Just by checking the subgroups generated by the elements concerned - no smart method - just working through the elements of the group checking the subgroups generated by each element. If the group concerned was very much bigger you would have to be more analytic I guess.

    Another possibility is <  r^2 > = {e,  r^2 } and of course there is {e}.

    Peter
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