---- > [!definition] Definition. ([[split short exact sequence]]) > A **split short exact sequence** is a [[short exact sequence]] $1 \to H \xrightarrow{\varphi} G \xrightarrow{\psi} K \to 1$ > such that $H$ admits a [[complementary subgroups|complement]] in $G$, namely, there is a [[subgroup]] $K' \leq G$ s.t. $K' \cap H = 1 \text{ and } G=HK'.$ A [[short exact sequence]] of $R$-[[module|modules]] $0 \to A \xrightarrow{f} B \xrightarrow{g} C \to 0$ **splits** if it is isomorphic to the [[short exact sequence]] $0 \to A \xrightarrow{\iota_{A}} A \oplus C \xrightarrow{\pi_{C}}C \to 0$ where $\iota_{A}$ is the canonical embedding and $\pi_{C}$ is the canonical projection. See also [[split short exact sequence of modules]], a bit of a duplicate. > [!basicexample] > Which of the following [[short exact sequence]]s are [[split short exact sequence]]s? (examples align with those in [[short exact sequence]]) ## $H=C_{2}, K=C_{2}$ — $G \in \{ C_{4}, C_{2} \times C_{2} \}$ ### $G \cong C_{4}$ **Not split**, because $C_{4}$ is not a [[external semi-direct product|semi-direct product]] of $C_{2}$ and $C_{2}$ for any [[group homomorphism|homomorphism]] $\phi$. ### $G \cong C_{2} \times C_{2}$ *Split*, because $C_{2} \times C_{2}$ is clearly a (semi-)[[direct product of groups|direct product]]. ## $H=C_{2}, K=C_{3}$ — $G \in \{ C_{2} \times C_{3} \cong C_{6}\}$ ### $G \cong C_{2} \times C_{3} \cong C_{6}$ *Split*, because $C_{2} \times C_{3} \cong C_{6}$ is clearly a (semi-)[[direct product of groups|direct product]]. ## $H=C_{3}, K=C_{2}$ — $G \in \{ C_{3} \times C_{2} \cong C_{6} , D_{3}\}$ ### $G \cong C_{3} \times C_{2} = C_{6}$ *Split*, because $C_{2} \times C_{3} \cong C_{6}$ is clearly a (semi-)[[direct product of groups|direct product]]. ### $G \cong D_{3}$ *Split*, because $D_{3} \cong C_{3} \rtimes C_{2}$. ## $H=C_{4}, K=C_{2}$ — $G \in \{C_{8}, C_{4} \times C_{2}, D_{4}, Q_{8}\}$ ### $G \cong C_{8}$ **Not split.** $1 \to C_{4} \to C_{8} \to C_{2} \to 1$. We would $C_{8}=C_{4} \rtimes_{\theta} C_{2}$ for some $\theta \in \hom(C_{2}, \text{Aut }C_{4})$. By [[internal semi-direct product]] this would mean there exist [[complementary subgroups]] of orders $4$ and $2$ in $C_{8}$, but from [[uniqueness of subgroups of finite cyclic groups]] we know that only one [[subgroup]] of order $4$ and of order $2$ exists in $C_{8}$ and the order 2 one, $\{ e, x^{4} \}$, is contained in the order 4 one, $\{ e,x^{2},x^{4},x^{6} \}$, so they cannot be [[complementary subgroups|complements]]. ### $G \cong C_{4} \times C_{2}$ *Split* as a [[direct product of groups]]. ### $G \cong D_{4}$ *Split* since $D_{4} \cong C_{4} \rtimes C_{2}$. ### $G \cong Q_{8}$ **Not split** because every [[subgroup]] of order $4$ contains $\{ \pm \b 1 \}$ so [[complementary subgroups]] cannot exist within $Q_{8}$. ## $H=C_{2}$, $K=C_{4}$ — $G \in \{ C_{8}, C_{2} \times C_{4} \}$ ### $G \cong C_{8}$ **Not split** because $C_{8}$ is not the product [[complementary subgroups]] and hence cannot be written as a [[external semi-direct product]]. ### $G \cong C_{2} \times C_{4}$ *Split* 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 > ```