Difference between revisions of "Derived series"
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A group <math>G</math> for which <math>D^n(G)</math> is [[trivial group |trivial]] for sufficiently large <math>n</math> is called [[solvable group |solvable]]. The least <math>n</math> such that <math>D^n(G) = \{ e\}</math> is called the ''solvability class'' of <math>G</math>. By transfinite recursion, this notion can be extended to infinite ordinals, as well. | A group <math>G</math> for which <math>D^n(G)</math> is [[trivial group |trivial]] for sufficiently large <math>n</math> is called [[solvable group |solvable]]. The least <math>n</math> such that <math>D^n(G) = \{ e\}</math> is called the ''solvability class'' of <math>G</math>. By transfinite recursion, this notion can be extended to infinite ordinals, as well. | ||
+ | |||
+ | Let <math>C^k(G)</math> be the <math>k</math>th term of the [[lower central series]] of <math>G</math>. Then from the relation <math>(C^m(G),C^n(G)) \subseteq C^{m+n}(G)</math> and induction, we have | ||
+ | <cmath> D^n(G) \subseteq C^{2^n}(G). </cmath> | ||
+ | In particular, if <math>G</math> is [[nilpotent group |nilpotent]] of class at most <math>2^n-1</math>, then it is solvable of class at most <math>n</math>. Thus if <math>G</math> is nilpotent, then it is solvable; however, the converse is not generally true. | ||
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 == |
Latest revision as of 00:13, 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.
Let be the
th term of the lower central series of
. Then from the relation
and induction, we have
In particular, if
is nilpotent of class at most
, then it is solvable of class at most
. Thus if
is nilpotent, then it is solvable; however, the converse is not generally true.
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
.