---- > [!definition] Definition. ([[normal series]]) > Let $G$ be a [[group]]. A **normal series** for $G$ is a nested [[sequence]] of [[subgroup|subgroups]] $G=G_{0} \trianglerighteq G_{1} \trianglerighteq \dots \trianglerighteq G_{r} =(e).$ > Thus each $G_{i+1}$ is [[normal subgroup|normal]] in $G_{i}$ (but not necessarily in the ambient [[group]] $G$.) > [!definition] Definition. (Equivalent normal series) > Two [[normal series]] $G=G_{0} \trianglerighteq G_{1} \trianglerighteq \dots \trianglerighteq G_{r}=(e)$ > and $G=H_{0} \trianglerighteq H_{1} \trianglerighteq \dots \trianglerighteq H_{s}=(e)$ > are said to be **equivalent** if the collection of successive [[quotient group|quotients]] $\{ G_{i} / G_{i+1} \}$ are [[group isomorphism|isomorphic]] to the successive [[quotient group|quotients]] $\{ H_{i} / H_{i+1} \}$ up to rearrangement. (In particular, we must have $r=s$.) ---- #### ---- #### 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 > ```