How to find information about a group given its presentation

**alsn** · November 16th 2011, 01:35 PM

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

**NonCommAlg** · November 16th 2011, 03:10 PM

Originally Posted by alsn

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 $G$ be your group and $g \in G.$ so $g = g_1g_2 \cdots g_n,$ for some $n$ , where each $g_i$ is either $x^{\pm 1}$ or $y^{\pm 1}.$ now, in $g$ we can replace $xy$ with $yx^{-1}$ and $x^{-1}y$ with $yx$ and so we will eventually get $g = y^ix^j$ for some $i,j.$ since $x^8=1$ and $y^2=x^4,$ we have $0 \leq i \leq 1$ and $0 \leq j \leq 7$ and thus $|G| \leq 16.$ finally, since $xy = yx^{-1},$ we have $x^my=yx^{-m}$ for all $m.$ use this to show that $g^8=1$ for all $g \in G$ and so $G$ is a $2$ -group.

**alsn** · November 17th 2011, 02:38 AM

thanks v much for the help!

**Swlabr** · November 17th 2011, 03:31 AM

Originally Posted by alsn

...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,

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

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