---- > [!definition] Definition. ([[Klein 4-group]]) > The [[group]] $C_{2} \times C_{2}$, called the **Klein 4-group**, is (up to [[group isomorphism|isomorphism]]) the smallest [[cyclic group|non-cyclic group]]. > > Denoting the elements of $C_{2} \times C_{2}$ using $Z_{2} \times Z_{2},$ the [[Cayley table]] is > | ⋅ | $(0, 0)$ | $(1, 0)$ | $(0, 1)$ | $(1, 1)$ | |-------------|-----------------------------------|-----------------------------------|-----------------------------------|-----------------------------------| | $(0, 0)$ | $\textcolor{SkyBlue}{(0, 0)}$ | $\textcolor{Red}{(1, 0)}$ | $\textcolor{Green}{(0, 1)}$ | $\textcolor{Purple}{(1, 1)}$ | | $(1, 0)$ | $\textcolor{Red}{(1, 0)}$ | $\textcolor{SkyBlue}{(0, 0)}$ | $\textcolor{Purple}{(1, 1)}$ | $\textcolor{Green}{(0, 1)}$ | | $(0, 1)$ | $\textcolor{Green}{(0, 1)}$ | $\textcolor{Purple}{(1, 1)}$ | $\textcolor{SkyBlue}{(0, 0)}$ | $\textcolor{Red}{(1, 0)}$ | | $(1, 1)$ | $\textcolor{Purple}{(1, 1)}$ | $\textcolor{Green}{(0, 1)}$ | $\textcolor{Red}{(1, 0)}$ | $\textcolor{SkyBlue}{(0, 0)}$ |. > > or, denoting the elements as $\{ e,a,b,c \}$: > > | ⋅ | $e$ | $a$ | $b$ | $c$ | |---------|-------------------------|-------------------------|-------------------------|-------------------------| | $e$ | $\textcolor{SkyBlue}{e}$ | $\textcolor{Red}{a}$ | $\textcolor{Green}{b}$ | $\textcolor{Purple}{c}$ | | $a$ | $\textcolor{Red}{a}$ | $\textcolor{SkyBlue}{e}$| $\textcolor{Purple}{c}$ | $\textcolor{Green}{b}$ | | $b$ | $\textcolor{Green}{b}$ | $\textcolor{Purple}{c}$ | $\textcolor{SkyBlue}{e}$| $\textcolor{Red}{a}$ | | $c$ | $\textcolor{Purple}{c}$ | $\textcolor{Green}{b}$ | $\textcolor{Red}{a}$ | $\textcolor{SkyBlue}{e}$| > Notice the $\textcolor{Skyblue}{\text{blue}}$ [[diagonal]]: $a^{2}=b^{2}=c^{2}=e$. > [!justification] > $C_{2} \times C_{2}$ is a [[group]] as a [[direct product of groups]]. ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```