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Math Help - <x,y:x=y=(xy)=1>

  1. #1
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    <x,y:x=y=(xy)=1>

    Can anybody help me in finding the all possible subgroups of the group
    <x,y:x=y=(xy)=1>. 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)=1>. Actually i want to find the the number of all subgroups of this group.

    Are you sure you meant the above or perhaps you meant to give a presentation of a group by generators and relators, in the

    form of G:= \langle x,y\;;\;x^2,y^3,(xy)^{12}\rangle ?

    Because if you meant what you wrote and not the above then we get:

    x^2=(xy)^{12}\Longrightarrow x=(yx)^{11}y\,,\,\,y^3=(xy)^{12}\Longrightarrow y^2=x(yx)^{11} , from where we get

    that your group can be presented as \langle a,y\;;\;y^2=aya\rangle\,,\,\,a=(yx)^{11} ... Check this.

    Tonio
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  3. #3
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    Isn't what you wrote the same thing? I thought those relations without equalities all were supposed to mean that those elements were equal to the identity.
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  4. #4
    MHF Contributor Swlabr's Avatar
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    Quote Originally Posted by topspin1617 View Post
    Isn't what you wrote the same thing? I thought those relations without equalities all were supposed to mean that those elements were equal to the identity.
    I don't know what he is doing either - certainly he has done something wrong in his calculations (his first group abelinises to C_2 \times C_3 while the second abelianises to \langle a, y; y=a^2 \rangle \cong \langle a; \emptyset \rangle = \mathbb{Z}, by Tietze transformations...)

    I suspect he didn't see the `=1' in your presentation, but even then your group abelianises to C_2 \times C_3. So...yeah, he's done something wrong...I have not idea where he gets his final presentation from.

    Anyway, to address your question, there isn't really a `neat' way of finding all the subgroups of a group that I know of. You just need to do it by hand. So, what is the order of your group? Which groups have order dividing your groups order?

    Next, take elements from your group and see what (sub)groups they generate. Then you want to write the subgroups out in a lattice, much like you can find here.
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