# Math Help - Group presentation question

1. ## Group presentation question

I have the group presentations

$\left\langle x,y|x^2=1,y^2=1,(xy)^n=1\right\rangle$

and

$\left\langle x,y|x^2=1,y^2=1\right\rangle$

and am told to say what well-known groups they define. I know the second is the infinite dihedral group and so am guessing the first is $D_n$, but don't know how to show why they are these groups.

Thanks

P.S. in an unrelated question, given a non-abelian group of order 60, how could you show that it has no odd permutations (without using the fact that it is $A_5$)?

2. ## Re: Group presentation question

Originally Posted by alsn
I have the group presentations

$\left\langle x,y|x^2=1,y^2=1,(xy)^n=1\right\rangle$

and

$\left\langle x,y|x^2=1,y^2=1\right\rangle$

and am told to say what well-known groups they define. I know the second is the infinite dihedral group and so am guessing the first is $D_n$, but don't know how to show why they are these groups.

Thanks

P.S. in an unrelated question, given a non-abelian group of order 60, how could you show that it has no odd permutations (without using the fact that it is $A_5$)?
Are you familiar with Tietze transformations? If so, try and get this presentation to look like the presentation of the Dihedral group which you should know,

$\langle a, b; a^2, aba^{-1}=b^{-1}, b^n\rangle$.

Otherwise, re-write the presentation you know so well so it "looks like" the presentation you are given. What should your isomorphism be? Then, prove it is an isomorphism!

3. ## Re: Group presentation question

i would suggest adding the generator u = xy, deriving from yu = yxy that yu = u^-1y (if x^2 = y^2 = 1, then (xy)^-1 = yx ,since (xy)(yx) = x(y^2)x = x^2 = 1),
and then re-writing u = xy as x = uy, so that we can eliminate x (because (uy)^2 = uyuy = uu^-1yy = 1).

all of that works for both groups.

4. ## Re: Group presentation question

Thanks Deveno, I did something similar but replacing x with zy^-1, then got that (zy^-1)^2=1 implies yzy^-1=z^-1 because y=y^-1, which gives the familiar presentation for D_n. i think this works too?