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


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

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

Consider the following inductive definition:


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?


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


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


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


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


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

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