Difference between revisions of "Derived series"
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By induction on <math>n</math> it follows that if <math>G</math> and <math>G'</math> are groups and <math>f : G \to G'</math> is a [[homomorphism]], then <math>f(D^n(G)) = D^n(f(G)) \subseteq D^n(G')</math>; in particular, if <math>f</math> is [[surjective]], <math>f(D^n(G)) = D^n(G')</math>. It follows that for all nonnegative integers <math>n</math>, <math>D^n(G)</math> is a [[characteristic subgroup]] of <math>G</math>. | By induction on <math>n</math> it follows that if <math>G</math> and <math>G'</math> are groups and <math>f : G \to G'</math> is a [[homomorphism]], then <math>f(D^n(G)) = D^n(f(G)) \subseteq D^n(G')</math>; in particular, if <math>f</math> is [[surjective]], <math>f(D^n(G)) = D^n(G')</math>. It follows that for all nonnegative integers <math>n</math>, <math>D^n(G)</math> is a [[characteristic subgroup]] of <math>G</math>. | ||
− | If <math>G=G_0, G_1, \dotsc</math> is a | + | If <math>G=G_0, G_1, \dotsc</math> is a decreasing sequence of subgroups such that <math>G_{k+1}</math> is a normal subgroup of <math>G_k</math> and <math>G_k/G_{k+1}</math> is [[abelian group |abelian]] for all integers <math>k</math>, then <math>D^k(G) \subseteq G_k</math>, by induction on <math>k</math>. |
== See also == | == See also == |
Revision as of 00:02, 2 June 2008
The derived series is a particular sequence of decreasing subgroups of a group .
Specifically, let be a group. The derived series is a sequence
defined recursively as
,
, where
is the derived group (i.e., the commutator subgroup) of a group
.
A group for which
is trivial for sufficiently large
is called solvable. The least
such that
is called the solvability class of
. By transfinite recursion, this notion can be extended to infinite ordinals, as well.
By induction on it follows that if
and
are groups and
is a homomorphism, then
; in particular, if
is surjective,
. It follows that for all nonnegative integers
,
is a characteristic subgroup of
.
If is a decreasing sequence of subgroups such that
is a normal subgroup of
and
is abelian for all integers
, then
, by induction on
.