---- > [!definition] Definition. ([[commutator subgroup]]) > Let $G$ be a [[group]]. The **commutator subgroup** or **derived subgroup** of $G$ is the ([[normal subgroup|normal]]) [[subgroup]] of $G$ that is [[generating set of a group|generated by]] all the [[commutator|commutators]] $[x,y]$, for all $x,y \in G$: $G':=\langle xy x ^{-1} y ^{-1} : x,y \in G \rangle .$ > \ > It is variously denoted by the symbols $G', G^{(1)},[G,G]$. > [!justification] > That $G'$ is [[normal subgroup|normal in]] $G$ can be concisely shown be observing that if $x$ lies in $G'$ and $g \in G$, then $gxg^{-1}=(gxg^{-1} x ^{-1}) \cdot x$ lies in $G'$ because $G'$ is closed under multiplication. > [!basicexample] > **$S_{3}.$** $[S_{3},S_{3}]=\{ e,\tau ,\tau^{2} \}=A_{3}$. This is merely because $A_{3}$ is the only [[normal subgroup]] of $S_{3}$. > \ > **$D_{4}$.** $[D_{4},D_{4}]=\{ e,x^{2} \}=Z(D_{4})$. To see this note that any interesting [[commutator]] of two elements has either the form $[x^{\ell }, x^{k}]=e$ or the form $[x^{\ell}, y]=x^{\ell}yx^{4-\ell }y=x^{ \ell}x^{-4 + \ell}=x^{2 \ell - 4} \in \{x^{2}, e\}$. And so any word in commutators is a word in $x^{2}$ or $e$ — so $[D_{4},D_{4}]=\{ e, x^{2} \}$. > \ > **$Q$.** $[Q,Q]=\{ \pm \v 1 \}$. Any interesting [[commutator]] of two elements has the form $[\b \ell, \v g]=\b \ell \v g \b \ell ^{-1} \v g ^{-1}=-\v g \b \ell \b \ell ^{-1} \v g ^{-1} \in \{ - \v 1 \}$, hence any word in [[commutator]]s is either $\v 1$ or $- \v 1$. > \ > $D_{5}.$ Repeat strategy for $D_{4}$, this time you'll get $\{ e,x,x^{2},x^{3},x^{4} \}.$ ---- #### ---- #### 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 > ```