The Klein 4-Group

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last updates: 12 December 2010

The Klein 4-Group

Let us make some shorter notations for the following matrices. $$\left(1,1\right)=\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right),\left(-1,1\right)=\left(\begin{array}{cc}-1& 0\\ 0& 1\end{array}\right),\left(1,-1\right)=\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right),\left(-1,-1\right)=\left(\begin{array}{cc}-1& 0\\ 0& -1\end{array}\right).$$

The Klein 4-group is the group of order $4$ defined as in the following table.

Set	Operation
$C_2\hspace{0.17em}\times \hspace{0.17em}{C}_2=\left\{\left(\begin{array}{cc}\pm 1& 0\\ 0& \pm 1\end{array}\right)\right\}$	Ordinary matrix multiplication

The compete multiplication table for this group is as follows.

Multiplication table
$\times$ $\left(1,1\right)$ $\left(1,-1\right)$ $\left(-1,1\right)$ $\left(-1,-1\right)$ $\left(1,1\right)$ $\left(1,1\right)$ $\left(1,-1\right)$ $\left(-1,1\right)$ $\left(-1,-1\right)$ $\left(1,-1\right)$ $\left(1,-1\right)$ $\left(1,1\right)$ $\left(-1,-1\right)$ $\left(-1,1\right)$ $\left(-1,-1\right)$ $\left(-1,-1\right)$ $\left(-1,1\right)$ $\left(1,-1\right)$ $\left(1,1\right)$

HW: Show that this group, as definded above, is isomorphic to the direct product of a cyclic group of order two, $C_2$, with another cyclic group of order two, $C_2$.

Center	Abelian	Conjugacy classes	Subgroups
$Z\left(G\right)=C_2\hspace{0.17em}\times \hspace{0.17em}{C}_2$	Yes	${𝒞}_{\left(1,1\right)}=\left\{\left(1,1\right)\right\}$	$H_0=C_2\hspace{0.17em}\times \hspace{0.17em}{C}_2$
		${𝒞}_{\left(1,-1\right)}=\left\{\left(1,-1\right)\right\}$	$H_1=\left\{\left(1,1\right),\left(1,-1\right)\right\}$
		${𝒞}_{\left(-1,1\right)}=\left\{\left(-1,1\right)\right\}$	$H_2=\left\{\left(1,1\right),\left(-1,1\right)\right\}$
		${𝒞}_{\left(-1,-1\right)}=\left\{\left(-1,-1\right)\right\}$	$H_3=\left\{\left(1,1\right),\left(\left(-1,-1\right)\right)\right\}$
			$H_4=\left\{\left(1,1\right)\right\}$

Element $g$	Order $o\left(g\right)$	Centralizer $Z_g$	Conjugacy Class ${𝒞}_g$
$\left(1,1\right)$	$1$	$C_2\hspace{0.17em}\times \hspace{0.17em}{C}_2$	${𝒞}_{\left(1,1\right)}$
$\left(1,-1\right)$	$2$	$C_2\hspace{0.17em}\times \hspace{0.17em}{C}_2$	${𝒞}_{\left(1,-1\right)}$
$\left(-1,1\right)$	$2$	$C_2\hspace{0.17em}\times \hspace{0.17em}{C}_2$	${𝒞}_{\left(-1,1\right)}$
$\left(-1,-1\right)$	$2$	$C_2\hspace{0.17em}\times \hspace{0.17em}{C}_2$	${𝒞}_{\left(-1,-1\right)}$

Generators	Relations
$x,y$	$x^2=1$
	$y^2=1$
	$xy=yx$

Subgroups $H_i$	Structure	Order $\left|H_i\right|$	Index	Normal	Quotient group
$H_0=C_2\hspace{0.17em}\times \hspace{0.17em}{C}_2$	$C_2\hspace{0.17em}\times \hspace{0.17em}{C}_2$	$4$	$1$	Yes	$(C_2\hspace{0.17em}\times \hspace{0.17em}{C}_2)/H_0\cong ⟨1⟩$
$H_1=\left\{\left(1,1\right),\left(1,-1\right)\right\}$	$C_2$	$2$	$2$	Yes	$(C_2\hspace{0.17em}\times \hspace{0.17em}{C}_2)/H_1\cong C_2$
$H_2=\left\{\left(1,1\right),\left(-1,1\right)\right\}$	$C_2$	$2$	$2$	Yes	$(C_2\hspace{0.17em}\times \hspace{0.17em}{C}_2)/H_2\cong C_2$
$H_3=\left\{\left(1,1\right),\left(-1,-1\right)\right\}$	$C_2$	$2$	$2$	Yes	$(C_2\hspace{0.17em}\times \hspace{0.17em}{C}_2)/H_3\cong C_2$
$H_4=\left\{\left(1,1\right)\right\}$	$⟨1⟩$	$1$	$4$	Yes	$(C_2\hspace{0.17em}\times \hspace{0.17em}{C}_2)/H_1\cong C_2\hspace{0.17em}\times \hspace{0.17em}{C}_2$

Subgroups $H_i$	Left Cosets	Right Cosets
$H_0$	$H_0=\left\{\left(\pm 1,\pm 1\right)\right\}$	$H_0=\left\{\left(\pm 1,\pm 1\right)\right\}$
$H_1$	$H_1=\left\{\left(1,1\right),\left(1,-1\right)\right\}$	$H_1=\left\{\left(1,1\right),\left(1,-1\right)\right\}$
	$\left(-1,1\right)H_1=\left\{\left(-1,1\right),\left(-1,-1\right)\right\}$	$H_1\left(-1,1\right)=\left\{\left(-1,1\right),\left(-1,-1\right)\right\}$
$H_2$	$H_2=\left\{\left(1,1\right),\left(-1,1\right)\right\}$	$H_2=\left\{\left(1,1\right),\left(-1,1\right)\right\}$
	$\left(1,-1\right)H_2=\left\{\left(1,-1\right),\left(-1,-1\right)\right\}$	$H_2\left(1,-1\right)=\left\{\left(1,-1\right),\left(-1,-1\right)\right\}$
$H_3$	$H_3=\left\{\left(1,1\right),\left(-1,-1\right)\right\}$	$H_3=\left\{\left(1,1\right),\left(-1,-1\right)\right\}$
	$\left(1,-1\right)H_3=\left\{\left(1,-1\right),\left(-1,1\right)\right\}$	$H_3\left(1,-1\right)=\left\{\left(1,-1\right),\left(-1,1\right)\right\}$
$H_4$	$H_4=\left\{\left(1,1\right)\right\}$	$H_4=\left\{\left(1,1\right)\right\}$
	$\left(-1,1\right)H_4=\left\{\left(-1,1\right)\right\}$	$H_4\left(-1,1\right)=\left\{\left(-1,1\right)\right\}$
	$\left(1,-1\right)H_4=\left\{\left(1,-1\right)\right\}$	$H_4\left(1,-1\right)=\left\{\left(1,-1\right)\right\}$
	$\left(-1,-1\right)H_4=\left\{\left(-1,-1\right)\right\}$	$H_4\left(-1,-1\right)=\left\{\left(-1,-1\right)\right\}$

Subgroups $H_i$	Normalizer $N_{H_i}$	Centralizer $Z_{H_i}$
$H_0$	$H_0$	$H_0$
$H_1$	$H_0$	$H_0$
$H_2$	$H_0$	$H_0$
$H_3$	$H_0$	$H_0$
$H_4$	$H_0$	$H_0$

Homomorphism	Kernel	Image
$\begin{array}{rrcc}{\phi }_0:& C_2\hspace{0.17em}\times \hspace{0.17em}{C}_2& \to & ⟨1⟩\\ & \left(-1,1\right)& \mapsto & 1\\ & \left(1,-1\right)& \mapsto & 1\end{array}$	$ker\hspace{0.17em}{\phi }_0=C_2\hspace{0.17em}\times \hspace{0.17em}{C}_2$	$im\hspace{0.17em}{\phi }_0=⟨1⟩$
$\begin{array}{rrcc}{\phi }_1& C_2\hspace{0.17em}\times \hspace{0.17em}{C}_2& \to & C_2\\ & \left(-1,1\right)& \mapsto & -1\\ & \left(1,-1\right)& \mapsto & 1\end{array}$	$ker\hspace{0.17em}{\phi }_1=H_1$	$im\hspace{0.17em}{\phi }_1=C_2$
$\begin{array}{rrcc}{\phi }_2& C_2\hspace{0.17em}\times \hspace{0.17em}{C}_2& \to & C_2\\ & \left(-1,1\right)& \mapsto & 1\\ & \left(1,-1\right)& \mapsto & -1\end{array}$	$ker\hspace{0.17em}{\phi }_2=H_2$	$im\hspace{0.17em}{\phi }_2=C_2$
$\begin{array}{rrcc}{\phi }_3& C_2\hspace{0.17em}\times \hspace{0.17em}{C}_2& \to & C_2\\ & \left(-1,1\right)& \mapsto & -1\\ & \left(1,-1\right)& \mapsto & -1\end{array}$	$ker\hspace{0.17em}{\phi }_3=H_3$	$im\hspace{0.17em}{\phi }_3=C_2$

References

[CM] H. S. M. Coxeter and W. O. J. Moser, Generators and relations for discrete groups, Fourth edition. Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], 14. Springer-Verlag, Berlin-New York, 1980. MR0562913 (81a:20001)

[GW1] F. Goodman and H. Wenzl, The Temperly-Lieb algebra at roots of unity, Pacific J. Math. 161 (1993), 307-334. MR1242201 (95c:16020)

page history