Isomorphism question

Apr 2010
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Let 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.
 
May 2010
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Let 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.
\(\displaystyle D_{2n}\) of order 2n have presentations

1. \(\displaystyle <r, s |r^n=1, s^2=1, sr=r^{-1}s>\),
2. \(\displaystyle <x, y | x^2 = y^2 = {(xy)}^n=1>\).

2 is denoted as \(\displaystyle I_2(n)\), implying that \(\displaystyle D_{2n}\) can also be generated by two reflections instead of one rotation and one reflection. You can find this idea in the context of Coxeter groups.

If you just play a little bit of your question, you will reduce your presentation to \(\displaystyle <a, b | a^2 = b^2 = {(ab)}^2=1>\), which is \(\displaystyle D_4\).

I'll leave it to you to covert the presentation 1 into 2 and verify that they are indeed equivalent.
 
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Apr 2010
22
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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.
 
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Apr 2010
22
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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>?
 
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May 2009
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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>?
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...

Alternatively, noting that you are in \(\displaystyle D_{2n}\) you can do the same trick. And yes, your group will have a center of order 2 in this case as \(\displaystyle D_{2n}\) has a center of order 2 if and only if \(\displaystyle n\) is even. Otherwise, it has trivial center.
 
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May 2010
95
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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.
In \(\displaystyle G/<y^2>\), simply plug \(\displaystyle y^2=1\) into your presentation of G.

Then, the original presentation reduces to \(\displaystyle <x, y|x^2 = y^2 = xyxy^{-1}=1>=<x, y|x^2 = y^2 = xyxy=1> \), where \(\displaystyle y=y^{-1}\) since \(\displaystyle y^2=1\). It is exactly the presentation of D_4 of order 4 (See my previous reply).

To convert the presentation 1 into presentation 2 in my previous reply, substitue s for x and sr for y. Then you will see that both are equivalent presentations.
 
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May 2009
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In \(\displaystyle G/<y^2>\), simply plug \(\displaystyle y^2=1\) into your presentation of G.
(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.

This is because, formally, \(\displaystyle \langle X ; R=1 \rangle :\cong F(X)/<<R>>\) where \(\displaystyle F(X)\) is the free group over the alphabet \(\displaystyle X\) and \(\displaystyle <<R>>\) is the normal subgroup generated by the elements of \(\displaystyle R\).)
 
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