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Abstract Algebra

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Higher Commutator Subgroup, Lower Central Series

ABSALG-FXJAKZ

For subgroups $H_1$, $H_2$ of a group $G$, denote by $[H_1,H_2]$ the subgroup of $G$ generated by the commutators:

$[h_1,h_2]$

...$h_1\in H_1$, $h_2\in H_2$.

Let $e$ be the neutral element of $G$.

Consider the following inductive definition:

$\gamma_1(G)=G$
$\gamma_2(G)=[\gamma_1(G),G]=[G,G]$
$\gamma_3(G)=[\gamma_2(G),G]=[[G,G],G]$
$\gamma_4(G)=[\gamma_3(G),G]=[[[G,G],G],G]$
$\gamma_{i+1}(G)=[\gamma_i(G),G]$

The subgroup $\gamma_i(G)$ of $G$ is called the $i$-th higher commutator subgroup of $G$.

The lower central series for $G$ is:

$$G=\gamma_1(G)\vartriangleright \gamma_2(G)\vartriangleright \gamma_3(G)\vartriangleright\ldots$$

The notation ${\mathcal D}_i(G)$ or $G_i$ is often used for $\gamma_i(G)$. If there is a smallest positive integer $n$ with $\gamma_{n+1}(G)=\{e\}$, then $n$ is called the length of the lower central series of $G$.

Which of the following is TRUE?

A

The lower central series of a perfect group has $\gamma_2(G)=G$ and $\gamma_i(G)=\{e\}$, $i\ge 3$.

B

The lower central series of an abelian group has $\gamma_2(G)=G$ and $\gamma_i(G)=\{e\}$, $i\ge 3$.

C

If $\gamma_n(G)=\{e\}$, for some finite integer $n$, then $G$ is soluble.

D

For every soluble group, we have $\gamma_n(G)=\{e\}$ for some finite integer $n$.

E

The lower central series of a group is the same as its derived series.