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Math Help - Regulat Octagon

  1. #1
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    Regulat Octagon

    Hi all

    Got this question but dont know where to start - need to get this in by tonight:

    The Group D8 (i.e. a regular Octagon with 8 sides) has 5 distinct sub-groups of order 4. What I need to do is find one cyclic sub group and one non-cyclic subgroup and also give their generators, or elements in standard form.

    Any ideas?

    Are the elements of the subgroup: {e, r, r^2, r^3, r^4, r^5, r^6, r^7, s, sr, sr^2, sr^3, sr^3, sr^4, sr^5, sr^6, sr^7}

    Please please help on this one - ive posted group theory and geometry questions here before but had little luck in answers these past few weeks.

    Cheers
    Muncha
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  2. #2
    MHF Contributor kalagota's Avatar
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    if you could post the elements of the group, maybe i could help.. did you mean D_8? This is the dihedral group on 8 vertices.
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  3. #3
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    Hi it is D8.

    The elements of D8 can be written as:
    Attached Thumbnails Attached Thumbnails Regulat Octagon-d8.jpg  
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  4. #4
    MHF Contributor kalagota's Avatar
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    im not sure about this.. let me recall my group theory.. you need a sub-group of order 4, which means a group with 4 elements right?

    consider \left\{e, r^4, s, sr^4\right\}.. i think this is isomorphic to Klein-4

    how about \left< r^2 \right> = \left\{e, r^2, r^4, r^6\right\}
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  5. #5
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    I think that is the element of the subgroup <r^4,s>.

    What I need is one cyclic and one non-cyclic sub group. Is the one you mentioned the non-cyclic group?
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  6. #6
    MHF Contributor kalagota's Avatar
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    the first set i posted is isomorphic to klein-4 and take note that klein-4 is not cyclic and hence the set cannot be cyclic..
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  7. #7
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    Thanks Kalagota

    So there are no cyclic groups, only a non cyclic group.

    Cheers for your help
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  8. #8
    MHF Contributor kalagota's Avatar
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    the second set i posted is a cyclic sub-group generated by \left< r^2\right>
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