Klein四元群的结构 Structure of Klein four-group

Klein四元群$K$是满足下列乘法表的群

Klein four-group is the group satisfying the following multpilicative table

$$\begin{array}{|c|c|c|c|c|}\hline    \cdot & 1  & a & b & c \\ \hline 1 & 1& a & b & c \\ \hline a & a & 1 & c & b \\ \hline b & b & c & 1 & a \\ \hline c & c & b & a & 1  \\ \hline \end{array}$$

显然，$K\cong \mathbb{Z}/2\times \mathbb{Z}/2$，这是最小的非循环群。

Clearly, $K\cong \mathbb{Z}/2\times \mathbb{Z}/2$, which is the smallest non-cyclic group.

我们知道如下三个事实

We know the following three facts

  • 毫无疑问，$K$的自同构群是第三个对称群$\mathfrak{S}_3$.

    There is no doubt that the automorphism group is the 3rd symmatric group $\mathfrak{S}_3$.

  • Klein四元群的特征表是

    The character table of Klein four group is

    $$\begin{array}{|c|c|c|c|c|}\hline  & 1 & a & b & c \\ \hline 1 & 1 & 1 & 1 & 1 \\ \hline \chi_a & 1 & 1 & -1 & -1 \\ \hline \chi_b & 1 & -1 & 1 & -1 \\ \hline \chi_c &1 & -1 & -1 & 1\\ \hline \end{array}$$

  • 最小的忠实置换表示是$\{1, (12)(34), (13)(24), (14)(23)\}\subseteq \mathfrak{S}_4$.

    The minimal faithful permutation representaion is $\{1, (12)(34), (13)(24), (14)(23)\}\subseteq \mathfrak{S}_4$.

上述皆可由由下述解释。考虑三维空间中以$(1,1,1), (1,-1,-1), (-1,1,-1),(-1,-,1,1)$为顶点正四面体，其中$a,b,c$在上面的作用分别是绕$x,y,z$轴旋转$180^\circ$. 这实际上给出所有正四面体的反射自同构。面的置换就是上面的忠实表示。

All of above can be summarized as follow. Consider the tetrahedron in 3D space whose vertices are $(1,1,1), (1,-1,-1), (-1,1,-1),(-1,-,1,1)$, where the action of $a$, $b$ and $c$ is the rotation of $180^\circ$ along axis $x$, $y$ and $z$ respectively. These actually give rise to all of the automorphisms of reflection of the tetrahedron. The permutation of faces is exactly the fait