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Derived series

From AoPSWiki

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 kth 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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