Derived series

The derived series is a particular sequence of decreasing subgroups of a group $G$.

Specifically, let $G$ be a group. The derived series is a sequence $(D^n(G))_{n \ge 0}$ defined recursively as $D^0(G)=G$, $D^{n+1}(G) = D(D^n(G))$, where $D(H)$ is the derived group (i.e., the commutator subgroup) of a group $H$.

A group $G$ for which $D^n(G)$ is trivial for sufficiently large $n$ is called solvable. The least $n$ such that $D^n(G) = \{ e\}$ is called the solvability class of $G$. By transfinite recursion, this notion can be extended to infinite ordinals, as well.

Let $C^k(G)$ be the $k$th term of the lower central series of $G$. Then from the relation $(C^m(G),C^n(G)) \subseteq C^{m+n}(G)$ and induction, we have \[D^n(G) \subseteq C^{2^n}(G).\] In particular, if $G$ is nilpotent of class at most $2^n-1$, then it is solvable of class at most $n$. Thus if $G$ is nilpotent, then it is solvable; however, the converse is not generally true.

By induction on $n$ it follows that if $G$ and $G'$ are groups and $f : G \to G'$ is a homomorphism, then $f(D^n(G)) = D^n(f(G)) \subseteq D^n(G')$; in particular, if $f$ is surjective, $f(D^n(G)) = D^n(G')$. It follows that for all nonnegative integers $n$, $D^n(G)$ is a characteristic subgroup of $G$.

If $G=G_0, G_1, \dotsc$ is a decreasing sequence of subgroups such that $G_{k+1}$ is a normal subgroup of $G_k$ and $G_k/G_{k+1}$ is abelian for all integers $k$, then $D^k(G) \subseteq G_k$, by induction on $k$.

See also

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