Grubun Cayley tablosu

Question

G bir grup ve  $$x,y \in G \ x^4 = 1 ,\ xy = yx^{-1}, \ x^2=y^2$$   ise bu grubun elemanlarını Cayley tablosunda gösteriniz. 

x ve y yi tek başına bırakıp değerini bulamıyorum, yardımcı olursanız sevinirim.

Answer 1

Oncelikle grubumuzun elemanlarini belirleyelim. Birim eleman grubun elemani olmali

Yani $G=\{1\}$ olabilir. Ama $x\in G$, o zaman $G=\{1,x\}$ dir. Ama  grup kapali olmali, yani $x\cdot x=x^2\in G$. O zaman $G=\{1,x,x^2\}$ dir. $G$ kapali olmali, yani $x\cdot x^2=x^3\in G$ olmali. $G=\{1,x,x^2,x^3\}$

$x\cdot x^3=x^4=1\in G$ , $x^2\cdot x^3=x^5=xx^4=x\in G$, $x^3\cdot x^3=x^6=x^2x^4=x^2\in G$ olmali, guzel duralim. Bir dakika, $y\in G$ demisler.

$G=\{1,x,x^2,x^3,y\}$ olur. Ama grup kapali, $\{xy,x^2y,x^3y\}$ ler de grubun elemani olmali.

$G=\{1,x,x^2,x^3,y,xy,x^2y,x^3y\}$ dir. ($y\cdot y=y^2=x^2$ verilmis.)

Oncelikle asagidaki tablonun asikar oldugunu soyleyelim ve $0$ lari grubun elemanlari olacak sekilde bulalim. Grubun her elemani her satirda ve sutunda bir defa bulunmali.

$$\begin{array}{c|cccccccc}  \cdot & 1 & x & x^2 & x^3 & y & xy & x^2y & x^3y \\\hline  1 & 1 & x & x^2 & x^3 & y & xy & x^2y & x^3y \\  x & x & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\  x^2 & x^2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\  x^3 & x^3 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\  y & y & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\  xy & xy & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\  x^2y & x^2y & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\  x^3y & x^3y & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \end{array}$$

2. adim

$(x)( x)=x^2$, $(x)( x^2)=x^3$ ve $x^4=1$ verilmis.

$$\begin{array}{c|cccccccc}  \cdot & 1 & x & x^2 & x^3 & y & xy & x^2y & x^3y \\\hline  1 & 1 & x & x^2 & x^3 & y & xy & x^2y & x^3y \\  x & x & x^2 & x^3 & 1 & 0 & 0 & 0 & 0 \\  x^2 & x^2 & x^3 & 1 & 0 & 0 & 0 & 0 & 0 \\  x^3 & x^3 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\  y & y & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\  xy & xy & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\  x^2y & x^2y & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\  x^3y & x^3y & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \end{array}$$

3. adim

$(x^3)(x^2)=x^5=x^4x=x$

$(x^2)(x^3)=x^5=x^4x=x$

$(x^3)(x^3)=x^6=x^4x^2=x^2$

$(y)(y)=y^2=x^2$ verilmis. 

$(xy)(y)=xy^2=xx^2=x^3$ olur.

$(x^2y)(y)=x^2y^2=x^2x^2=x^4=1$ olur.

$(x^3y)(y)=x^3y^2=x^3x^2=x^5=xx^4=x$ olur.

$$\begin{array}{c|cccccccc}  \cdot & 1 & x & x^2 & x^3 & y & xy & x^2y & x^3y \\\hline  1 & 1 & x & x^2 & x^3 & y & xy & x^2y & x^3y \\  x & x & x^2 & x^3 & 1 & 0 & 0 & 0 & 0 \\  x^2 & x^2 & x^3 & 1 & x & 0 & 0 & 0 & 0 \\  x^3 & x^3 & 1 & x & x^2 & 0 & 0 & 0 & 0 \\  y & y & 0 & 0 & 0 & x^2 & 0 & 0 & 0 \\  xy & xy & 0 & 0 & 0 & x^3 & 0 & 0 & 0 \\  x^2y & x^2y & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\  x^3y & x^3y & 0 & 0 & 0 & x & 0 & 0 & 0 \\ \end{array}$$

4. adim

$xy$  ve  $yx$ 'e  ihtiyacimiz var (grub degismeli olmayabilir). $x^4=1\Longrightarrow x^4x^{-1}=x^{-1}\Longrightarrow x^3=x^{-1}$ (bu tablodan da gorulebilir, $xz=1$ olacak sekilde bir $z$ bul)

• $(x)(y)=xy$
  

$xy=yx^{-1}$ verilmis, $xy=yx^3$ olur.

$xy=yx^3\Longrightarrow y=x^{-1}yx^3\Longrightarrow yx=x^{-1}yx^4\Longrightarrow yx=x^3y$ olur.

• $(y)(x)=x^3y$

$yx=x^3y\Longrightarrow yx^2=x^3yx\Longrightarrow yx^2=x^3x^3y=x^4x^2y=x^2y$

• $(y)(x^2)=x^2y$

$yx=x^3y\Longrightarrow yx^2=x^3yx\Longrightarrow yx^2=x^3x^3y\Longrightarrow yx^3=x^3x^3yx\Longrightarrow yx^3=x^3x^3x^3y=x^4x^4xy=xy$

• $(y)(x^3)=xy$

$xy=yx^3\Longrightarrow y(xy)=y^2x^3=x^2x^3=x$

$(x^2)(y)=y^2y=y^3=yy^2=yx^2\Longrightarrow$

• $(x^2)(y)=x^2y$

$yx=x^3y\Longrightarrow xyx=x^4y=y$

• $(xy)x=y$

$$\begin{array}{c|cccccccc}  \cdot & 1 & x & x^2 & x^3 & y & xy & x^2y & x^3y \\\hline  1 & 1 & x & x^2 & x^3 & y & xy & x^2y & x^3y \\  x & x & x^2 & x^3 & 1 & xy& 0 & 0 & 0 \\  x^2 & x^2 & x^3 & 1 & x & x^2y & 0 & 0 & 0 \\  x^3 & x^3 & 1 & x & x^2 & 0 & 0 & 0 & 0 \\  y & y & x^3y & x^2y & xy & x^2 & x & 0 & 0 \\  xy & xy & y & 0 & 0 & x^3 & 0 & 0 & 0 \\  x^2y & x^2y & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\  x^3y & x^3y & 0 & 0 & 0 & x & 0 & 0 & 0 \\ \end{array}$$

5. adim

$(xy)(xy)=x(yx)y=x(x^3y)y=x^4y^2=y^2=x^2$

• $(xy)(xy)=x^2$

$(x^2y)(x)=x^2(yx)=x^2(x^3y)=x^5y=xy$

• $(x^2y)(x)=xy$

$(x^2y)(x^2)=x^2(yx)x=x^2(x^3y)x=xyx=yx=xx^3y=x^4y=y$

• $(x^2y)(x^2)=y$

$(x^2y)(x^3)=x^2(yx)x^2=x^2(x^3y)x^2=xyx^2=xyy^2=xy^3=xy^2y=xx^2y=x^3y$

• $(x^2y)(x^3)=x^3y$

$(x^3y)(x)=x^3(yx)=x^3(x^3y)=x^2y$

• $(x^3y)(x)=x^2y$

$(x^3y)(x^2)=x^3(yx)x=x^3(x^3y)x=x^2yx=x^2x^3y=xy$

• $(x^3y)(x^2)=xy$

$(x^3y)(x^3)=x^3(yx)x^2=x^3(x^3y)x^2=x^2yx^2=y^2yy^2=y^5=y^4y=x^4y=y$

• $(x^3y)(x^3)=y$

$$\begin{array}{c|cccccccc}  \cdot & 1 & x & x^2 & x^3 & y & xy & x^2y & x^3y \\\hline  1 & 1 & x & x^2 & x^3 & y & xy & x^2y & x^3y \\  x & x & x^2 & x^3 & 1 & xy & 0 & 0 & 0 \\  x^2 & x^2 & x^3 & 1 & x & x^2y & 0 & 0 & 0 \\  x^3 & x^3 & 1 & x & x^2 & 0 & 0 & 0 & 0 \\  y & y & x^3y & x^2y & xy & x^2 & x & 0 & 0 \\  xy & xy & y & 0 & 0 & x^3 & x^2 & 0 & 0 \\  x^2y & x^2y & xy & y & x^3y & 1 & 0 & 0 & 0 \\  x^3y & x^3y & x^2y & xy & y & x & 0 & 0 & 0 \\ \end{array}$$

Answer 2

6. adim

• $(x)(xy)=x^2y$
• $(x)(x^2y)=x^3y$
• $(x)(x^3y)=x^4y=y$
• $(x^2)(xy)=x^3y$
• $(x^2)(x^2y)=y$
• $(x^2)(x^3y)=x^4xy=xy$
• $(x^3)(xy)=y$
• $(x^3)(x^2y)=xy$
• $(x^3)(x^3y)=x^2y$

$$\begin{array}{c|cccccccc}  \cdot & 1 & x & x^2 & x^3 & y & xy & x^2y & x^3y \\\hline  1 & 1 & x & x^2 & x^3 & y & xy & x^2y & x^3y \\  x & x & x^2 & x^3 & 1 & xy & x^2y & x^3y & y \\  x^2 & x^2 & x^3 & 1 & x & x^2y & x^3y & y & xy \\  x^3 & x^3 & 1 & x & x^2 & x^3y & y & xy & x^2y \\  y & y & x^3y & x^2y & xy & x^2 & x & 0 & 0 \\  xy & xy & y & 0 & 0 & x^3 & x^2 & 0 & 0 \\  x^2y & x^2y & xy & y & x^3y & 1 & 0 & 0 & 0 \\  x^3y & x^3y & x^2y & xy & y & x & 0 & 0 & 0 \\ \end{array}$$

Devamini getirirsiniz.. Eger satir veya sutunda bir elaman eksikse, doldurmak kolay, grubun hangi elemani o satir veya sutunda yoksa, son eleman o elamandir.

Mesela $(x^3)(y)=x^3y$ oldugunu islem yapmadan soyleyebiliriz, cunku $y$ sutununda yalnizca o eleman eksik.

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Teşekkürlerimi sunuyorum hocam.