# Group presentation question

• Nov 22nd 2011, 01:28 AM
alsn
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 $A_5$)?
• Nov 22nd 2011, 01:58 AM
Swlabr
Re: Group presentation question
Quote:

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!