---- > [!definition] Definition. ([[solvable group]]) > A [[group]] $G$ is **solvable** if it admits a [[normal series]] in which every successive [[quotient group|quotient]] is [[abelian group|abelian]]. > > [!equivalence] > The following are equivalent. > - $G$ is solvable; > - The [[commutator series]] of $G$ is a [[normal series]]; > - The [[commutator series]] of $G$ terminates at $(e)$, i.e., $G^{(n)}=(e)$ for some $n$. > > [!proof] Proof of Equivalence. > [!basicnonexample] Remark. > To see why it isn't always true that the [[commutator series|derived series]] of $G$ is a [[normal series]], consider the [[commutator series of the symmetric group|derived series of]] $S_{5}$: $S_{5} \geq S_{5}^{(1)}=A_{5}$ which does not terminate at $(e)$ because $A_{5}$ is [[simple group|simple]] and non[[abelian group|abelian]]. $(1 \implies 3)$. Suppose $G$ is [[solvable group|solvable]], i.e., it admits a [[normal series]] in which every successive [[quotient group|quotient]] is [[abelian group|abelian]]. So we can find $G=G_{0} \trianglerighteq G_{1} \trianglerighteq \dots \trianglerighteq G_{n}=(e)$ such that $G_{i+1} / G_{i}$ is [[abelian group|abelian]]. Then by [[universal property of commutator subgroup]] we have $G_{i} \geq G^{(i)}$ for all $i$. In particular, $(e)=G_{n} \geq G_{}^{(n)}$ and $(3)$ is satisfied. $(3 \implies 1)$. $G / [G,G]$ is [[abelian group|abelian]] for any $G$, that is, successive quotients in the [[commutator series]] of $G$ are [[abelian group|abelian]]. Since the [[commutator series]] terminates at $(e)$, it is in fact a [[normal series]]. Suppose $N \trianglelefteq G$. Then $[N,N]$ is [[normal subgroup|normal]] not just in $N$ but also in $G$. $gN'g^{-1}$ $g(n_{1}n_{2}n_{1}^{-1}n_{2}^{-1})g^{-1}$ $=gn_{1} g^{-1} g n_{2} g^{-1} gn_{1}^{-1} g^{-1} g n_{2} ^{-1}g^{-1}$ $=gn_{1}g^{-1} \ \ gn_{2}g^{-1} \ \ (gn_{1} g^{-1})^{-1} \ \ (gn_{2} g ^{-1})^{-1}$ $\in N'$ since $N$ is [[normal subgroup]]. ---- #### ---- #### 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 > ```