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Thread: How to find information about a group given its presentation

  1. #1
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    How to find information about a group given its presentation

    hi,

    i am given the group presentation

    <x,y | x^8=1, x^4=y^2, xy=yx^{-1}>

    and am told to prove that it defines a 2-group of order at most 16. i've been playing around with the relations but not really sure how to go about this. thought i might have to see how many different elements can be made and see that it's at most 16 but don't know about the 2-group part. in general i'm not sure how to go about finding information about a group knowing its presentation. any help?

    thank youuu

    xxx
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  2. #2
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    Re: How to find information about a group given its presentation

    Quote Originally Posted by alsn View Post
    hi,

    i am given the group presentation

    <x,y | x^8=1, x^4=y^2, xy=yx^{-1}>

    and am told to prove that it defines a 2-group of order at most 16. i've been playing around with the relations but not really sure how to go about this. thought i might have to see how many different elements can be made and see that it's at most 16 but don't know about the 2-group part. in general i'm not sure how to go about finding information about a group knowing its presentation. any help?

    thank youuu

    xxx
    let $\displaystyle G$ be your group and $\displaystyle g \in G.$ so $\displaystyle g = g_1g_2 \cdots g_n,$ for some $\displaystyle n$, where each $\displaystyle g_i$ is either $\displaystyle x^{\pm 1}$ or $\displaystyle y^{\pm 1}.$ now, in $\displaystyle g$ we can replace $\displaystyle xy$ with $\displaystyle yx^{-1}$ and $\displaystyle x^{-1}y$ with $\displaystyle yx$ and so we will eventually get $\displaystyle g = y^ix^j$ for some $\displaystyle i,j.$ since $\displaystyle x^8=1$ and $\displaystyle y^2=x^4,$ we have $\displaystyle 0 \leq i \leq 1$ and $\displaystyle 0 \leq j \leq 7$ and thus $\displaystyle |G| \leq 16.$ finally, since $\displaystyle xy = yx^{-1},$ we have $\displaystyle x^my=yx^{-m}$ for all $\displaystyle m.$ use this to show that $\displaystyle g^8=1$ for all $\displaystyle g \in G$ and so $\displaystyle G$ is a $\displaystyle 2$-group.
    Last edited by NonCommAlg; Nov 17th 2011 at 09:42 AM.
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  3. #3
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    Re: How to find information about a group given its presentation

    thanks v much for the help!
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  4. #4
    MHF Contributor Swlabr's Avatar
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    Re: How to find information about a group given its presentation

    Quote Originally Posted by alsn View Post
    ...in general i'm not sure how to go about finding information about a group knowing its presentation. any help?
    This is a good question, but with a somewhat rubbishy answer: No one does. Not really. I mean, there are some presentations where you cannot tell if a given element is equal to the identity or not! Look up "the word problem for groups".

    Also, there is a famous example for the 60s or so, where John Conway posed a problem in Notices to prove that a certain group was cyclic of order 15, I believe. It took something like two years for the solution to be found! Look up Fibonacci groups for more details. I think the group was,

    $\displaystyle \langle x_1, x_2, x_3, x_4, x_5; x_1{x_2}=x_3, x_2{x_3}=x_4, x_3{x_4}=x_5, x_4{x_5}=x_1, x_5{x_1}=x_2\rangle$.
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