The Klein 4-Group

## The Klein 4-Group

Let us make some shorter notations for the following matrices. $1,1 = 1 0 0 1 , -1,1 = -1 0 0 1 , 1,-1 = 1 0 0 -1 , -1,-1 = -1 0 0 -1 .$

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

Set Operation
${C}_{2} × {C}_{2}=\left\{\left(\begin{array}{cc}±1& 0\\ 0& ±1\end{array}\right)\right\}$ Ordinary matrix multiplication

The compete multiplication table for this group is as follows.

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

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
$Z\left(G\right)={C}_{2} × {C}_{2}$ Yes ${𝒞}_{\left(1,1\right)}=\left\{\left(1,1\right)\right\}$ ${H}_{0}={C}_{2} × {C}_{2}$
${𝒞}_{\left(1,-1\right)}=\left\{\left(1,-1\right)\right\}$ ${H}_{1}=\left\{\left(1,1\right),\left(1,-1\right)\right\}$
${𝒞}_{\left(-1,1\right)}=\left\{\left(-1,1\right)\right\}$ ${H}_{2}=\left\{\left(1,1\right),\left(-1,1\right)\right\}$
${𝒞}_{\left(-1,-1\right)}=\left\{\left(-1,-1\right)\right\}$ ${H}_{3}=\left\{\left(1,1\right),\left(\left(-1,-1\right)\right)\right\}$
${H}_{4}=\left\{\left(1,1\right)\right\}$

Element $g$ Order $ο\left(g\right)$ Centralizer ${Z}_{g}$ Conjugacy Class ${𝒞}_{g}$
$\left(1,1\right)$ $1$ ${C}_{2} × {C}_{2}$ ${𝒞}_{\left(1,1\right)}$
$\left(1,-1\right)$ $2$ ${C}_{2} × {C}_{2}$ ${𝒞}_{\left(1,-1\right)}$
$\left(-1,1\right)$ $2$ ${C}_{2} × {C}_{2}$ ${𝒞}_{\left(-1,1\right)}$
$\left(-1,-1\right)$ $2$ ${C}_{2} × {C}_{2}$ ${𝒞}_{\left(-1,-1\right)}$

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
${H}_{0}={C}_{2} × {C}_{2}$ ${C}_{2} × {C}_{2}$ $4$ $1$ Yes $\left({C}_{2} × {C}_{2}\right)/{H}_{0}\cong ⟨1⟩$
${H}_{1}=\left\{\left(1,1\right),\left(1,-1\right)\right\}$ ${C}_{2}$ $2$ $2$ Yes $\left({C}_{2} × {C}_{2}\right)/{H}_{1}\cong {C}_{2}$
${H}_{2}=\left\{\left(1,1\right),\left(-1,1\right)\right\}$ ${C}_{2}$ $2$ $2$ Yes $\left({C}_{2} × {C}_{2}\right)/{H}_{2}\cong {C}_{2}$
${H}_{3}=\left\{\left(1,1\right),\left(-1,-1\right)\right\}$ ${C}_{2}$ $2$ $2$ Yes $\left({C}_{2} × {C}_{2}\right)/{H}_{3}\cong {C}_{2}$
${H}_{4}=\left\{\left(1,1\right)\right\}$ $⟨1⟩$ $1$ $4$ Yes $\left({C}_{2} × {C}_{2}\right)/{H}_{1}\cong {C}_{2} × {C}_{2}$

Subgroups ${H}_{i}$ Left Cosets Right Cosets
${H}_{0}$ ${H}_{0}=\left\{\left(±1,±1\right)\right\}$ ${H}_{0}=\left\{\left(±1,±1\right)\right\}$
${H}_{1}$ ${H}_{1}=\left\{\left(1,1\right),\left(1,-1\right)\right\}$ ${H}_{1}=\left\{\left(1,1\right),\left(1,-1\right)\right\}$
$\left(-1,1\right){H}_{1}=\left\{\left(-1,1\right),\left(-1,-1\right)\right\}$ ${H}_{1}\left(-1,1\right)=\left\{\left(-1,1\right),\left(-1,-1\right)\right\}$
${H}_{2}$ ${H}_{2}=\left\{\left(1,1\right),\left(-1,1\right)\right\}$ ${H}_{2}=\left\{\left(1,1\right),\left(-1,1\right)\right\}$
$\left(1,-1\right){H}_{2}=\left\{\left(1,-1\right),\left(-1,-1\right)\right\}$ ${H}_{2}\left(1,-1\right)=\left\{\left(1,-1\right),\left(-1,-1\right)\right\}$
${H}_{3}$ ${H}_{3}=\left\{\left(1,1\right),\left(-1,-1\right)\right\}$ ${H}_{3}=\left\{\left(1,1\right),\left(-1,-1\right)\right\}$
$\left(1,-1\right){H}_{3}=\left\{\left(1,-1\right),\left(-1,1\right)\right\}$ ${H}_{3}\left(1,-1\right)=\left\{\left(1,-1\right),\left(-1,1\right)\right\}$
${H}_{4}$ ${H}_{4}=\left\{\left(1,1\right)\right\}$ ${H}_{4}=\left\{\left(1,1\right)\right\}$
$\left(-1,1\right){H}_{4}=\left\{\left(-1,1\right)\right\}$ ${H}_{4}\left(-1,1\right)=\left\{\left(-1,1\right)\right\}$
$\left(1,-1\right){H}_{4}=\left\{\left(1,-1\right)\right\}$ ${H}_{4}\left(1,-1\right)=\left\{\left(1,-1\right)\right\}$
$\left(-1,-1\right){H}_{4}=\left\{\left(-1,-1\right)\right\}$ ${H}_{4}\left(-1,-1\right)=\left\{\left(-1,-1\right)\right\}$

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
$\begin{array}{rrcc}{\phi }_{0}:& {C}_{2} × {C}_{2}& \to & ⟨1⟩\\ & \left(-1,1\right)& ↦& 1\\ & \left(1,-1\right)& ↦& 1\end{array}$ $ker {\phi }_{0}={C}_{2} × {C}_{2}$ $im {\phi }_{0}=⟨1⟩$
$\begin{array}{rrcc}{\phi }_{1}& {C}_{2} × {C}_{2}& \to & {C}_{2}\\ & \left(-1,1\right)& ↦& -1\\ & \left(1,-1\right)& ↦& 1\end{array}$ $ker {\phi }_{1}={H}_{1}$ $im {\phi }_{1}={C}_{2}$
$\begin{array}{rrcc}{\phi }_{2}& {C}_{2} × {C}_{2}& \to & {C}_{2}\\ & \left(-1,1\right)& ↦& 1\\ & \left(1,-1\right)& ↦& -1\end{array}$ $ker {\phi }_{2}={H}_{2}$ $im {\phi }_{2}={C}_{2}$
$\begin{array}{rrcc}{\phi }_{3}& {C}_{2} × {C}_{2}& \to & {C}_{2}\\ & \left(-1,1\right)& ↦& -1\\ & \left(1,-1\right)& ↦& -1\end{array}$ $ker {\phi }_{3}={H}_{3}$ $im {\phi }_{3}={C}_{2}$

## 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)