---- > [!definition] Definition. ([[commutator series]]) > Iteratively take [[commutator subgroup|commutators]] of a [[group]] $G$. The **commutator series** or **derived series** of $G$ is $G = G^{(0)} \geq G^{(1)} \geq \dots $ > [!basicexample] Examples. > - [[commutator series of order-12 groups]] > - [[commutator series of the symmetric group]] ---- #### ---- #### 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 > ```