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Math Help - subgroups of the group <x,y:x=y=(xy)>

  1. #1
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    subgroups of the group <x,y:x=y=(xy)>

    Can anybody help me in finding the all possible subgroups of the group
    <x,y:x=y=(xy)>. Actually i want to find the the number of all subgroups of this group.
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  2. #2
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    Quote Originally Posted by makenqau View Post
    Can anybody help me in finding the all possible subgroups of the group
    <x,y:x=y=(xy)>. Actually i want to find the the number of all subgroups of this group.

    x^2=(xy)^{12}\Longrightarrow x=y(xy)^{11}\,,\,and\,\,also\,\,y^3=(xy)^{12}\Long  rightarrow y^2=(xy)^{11}x=(xy)^{11}y(xy)^{11} , thus it

    follows that \langle x,y\;;\;x^2=y^3=(xy)^{12}\rangle =\langle {y,(xy)^{11}\rangle \cong F_2= the free group in two (free) generators.

    Now, did you actually mean to write what you did or you meant to give a presentation of the group in the

    form \langle x,y\;;\;x^2,y^3,(xy)^{12}\rangle ? Because this last is NOT a free group as it has non-trivial torsion elements, so

    we get two different groups...or I'm wrong, of course. Check this.

    Tonio
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