\(\displaystyle D_{2n}\) of order 2n have presentationsLet G=<x, y|x^4=y^4=e, xyxy^–1=e>. Show that |G|≤16. Assuming |G|=16, show G/<y^2> is isomorphic to D4.
Looking at the presentation you give, the third relation gives you something which looks like commutativity. Essentially, it allows you to write every element in the form \(\displaystyle x^iy^j\) for \(\displaystyle i, j \in \{0, 1, 2, 3\}\). What you need to do is find the values of \(\displaystyle i,j\) such that \(\displaystyle x^iy^jx^ay^b = x^ay^ba^iy^j\) for all \(\displaystyle a, b\). So just plug it in...how would I find the center for this group if the groups order was 16? I know that since xyxy^-1=e that xy=yx^3, so x=yx^3y^-1, so than x^2y=xyx^3=yx^6=yx^2. So x^2 and e is in the center. How would I show that nothing else is in the center besides <x^2>?
In \(\displaystyle G/<y^2>\), simply plug \(\displaystyle y^2=1\) into your presentation of G.Here is what I have:
since xyxy^-1=e, we know that yxy^-1=x^-1=x^3, so we know that <x) is normal. So G= <x> union y<x> union y^2<x> union y^3<x> and |G|<= 16.
Lets assume that |G|=16.
So |G|/|<y^2>|=8, thus |G|/|<y^2>|>= |D4|.
D4=<a,b|a^4=b^2=(ab)^2=e, or ab=ba^3>.
Let w=x<y^2> , z=y<y^2> and q=e<y^2> (where q is are identity element in G/<y^2>.
Need to show that G/<y^2>= <w,z|w^4=z^2=(wz)^2=e>.
1. Since x and y are generates in G, then w and z are generates in G/<y^2>. I am not sure if this is right, could someone explain this to me? (I need to show that w and z generate G/<y^2>).
2. w^4=e since (x<y^2>)^4=x^4<y^2>=e<y^2>=q.
3. z^2=e since (y<y^2>)^2=y^2<y^2>=e<y^2>=q.
4. (wz)^2=(wz)(wz)=e. Since wz=(z^-1)(w^-1), then wz=xy<y^2> and z^-1w^-1=y^3x^3<y^2>=yx^3<y^2>. Thus xy<y^2>=yx^3<y^2>.
Therefore, G/<y^2> is isomorphic to D4.
(It should perhaps be pointed out that relations in a presentation, homomorphic images and quotients are all equivalent. Thus, putting a relation into your presentation is the same as taking a quotient.In \(\displaystyle G/<y^2>\), simply plug \(\displaystyle y^2=1\) into your presentation of G.
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