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

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

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

thanks v much for the help!

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

Quote:

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,

.