Fitting length: Difference between revisions

Latest revision as of 22:22, 8 August 2012

This article defines an arithmetic function on a restricted class of groups, namely: solvable groups

Definition

Given a solvable group, we can define its Fitting length or nilpotent length in the following equivalent ways:

It is the minimum possible length of a Fitting series (also called a nilpotent series) for the group, where a Fitting series is a subnormal series where all the successive quotients are nilpotent groups.
If the group is a finite solvable group: It is the length of the upper Fitting series of the group. The upper Fitting series is an ascending series defined via quotient-iteration of the Fitting subgroup, which is the join of all nilpotent normal subgroups. That is, we define $\operatorname {Fit} _{0}(G)$ as the trivial subgroup $\operatorname {Fit} _{i}(G)$ so that $\operatorname {Fit} _{i}(G)/\operatorname {Fit} _{i-1}(G)$ is the Fitting subgroup of $G/\operatorname {Fit} _{i-1}(G)$ .
If the group is a finite solvable group: It is the length of the lower Fitting series of the group. The lower Fitting series is a descending series defined via iteration of the nilpotent residual, where the nilpotent residual is the intersection of all normal subgroups with nilpotent quotients. For finite groups, the nilpotent residual coincides with the hypocenter.

For an infinite group, a join of nilpotent normal subgroups need not be nilpotent and an intersection of nilpotent-quotient normal subgroups need not have nilpotent quotient. In both cases, the problem is that the nilpotency class may become unbounded. Although we can still define the Fitting subgroup and the nilpotent residual, the Fitting subgroup need not be nilpotent and the nilpotent residual need not have nilpotent quotient.

Facts

The Fitting length of a solvable group is 1 if and only if it is nilpotent.
The Fitting length of any solvable group is bounded from above by its derived length.

@@ Line 3: / Line 3: @@
 ==Definition==
-===Symbol-free definition===
+Given a [[solvable group]], we can define its '''Fitting length''' or '''nilpotent length''' in the following equivalent ways:
-Given a [[solvable group]], we can define its '''Fitting length''' as follows: It is the length of the [[Fitting series]] of the group. The Fitting series is an ascending series defined via quotient-iteration of the [[Fitting subgroup]]. That is, we define <math>F_i(G)</math> so that <math>F_i(G)/F_{i-1}</math> is the Fitting subgroup of <math>G/F_{i-1}(G)</math>.
+# It is the minimum possible length of a [[Fitting series]] (also called a ''nilpotent series'') for the group, where a Fitting series is a [[subnormal series]] where all the successive quotients are [[nilpotent group]]s.
+# If the group is a [[finite solvable group]]: It is the length of the [[defining ingredient::upper Fitting series]] of the group. The upper Fitting series is an ascending series defined via quotient-iteration of the [[defining ingredient::Fitting subgroup]], which is the join of all [[nilpotent normal subgroup]]s. That is, we define <math>\operatorname{Fit}_0(G)</math> as the trivial subgroup <math>\operatorname{Fit}_i(G)</math> so that <math>\operatorname{Fit}_i(G)/\operatorname{Fit}_{i-1}(G)</math> is the Fitting subgroup of <math>G/\operatorname{Fit}_{i-1}(G)</math>.
+# If the group is a [[finite solvable group]]: It is the length of the [[defining ingredient::lower Fitting series]] of the group. The lower Fitting series is a descending series defined via iteration of the [[defining ingredient::nilpotent residual]], where the nilpotent residual is the intersection of all [[normal subgroup]]s with nilpotent quotients. For finite groups, the nilpotent residual coincides with the [[hypocenter]].
+For an infinite group, a join of nilpotent normal subgroups need not be nilpotent and an intersection of [[nilpotent-quotient normal subgroup]]s need not have nilpotent quotient. In both cases, the problem is that the nilpotency class may become unbounded. Although we can still define the Fitting subgroup and the nilpotent residual, the Fitting subgroup need not be nilpotent and the nilpotent residual need not have nilpotent quotient.
+==Facts==
+* The [[Fitting length]] of a solvable group is 1 if and only if it is nilpotent.
+* The [[Fitting length]] of any solvable group is bounded from above by its [[derived length]].