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Derived length

, 15:56, 24 June 2009
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===Symbol-free definition===
Given a [[solvable group]], we define its '''solvable derived length''' or '''derived solvable length''' as follows:
* It is the length of the [[derived series]] of the group. Note here that by length of the series, we mean the number of successive inclusions, so the length is one less than the actual number of subgroups in the derived series.
* It is the minimum possible length of a [[subnormal series]] from the trivial subgroup to the whole group such that all the quotients in the series are [[abelian group]]s.
When we say that a group has solvable derived length [itex]k[/itex], we typically mean that it has solvable length at most [itex]k[/itex].
==Facts==
===Small values===
* A group has solvable derived length zero if and only if it is [[trivial group|trivial]].* A group has solvable derived length at most one if and only if it is [[abelian group|abelian]].* A group has solvable derived length at most two if and only if it is a [[metabelian group]]: it has an [[abelian normal subgroup]] such that the [[quotient group]] is also an [[abelian group]].===Relation with nilpotence class===
===Relation with nilpotency class=== {{further|[[Nilpotence Nilpotency class versus solvable length]]}}
Any nilpotent group is solvable, and there are numerical relations between the nilpotence class and solvable length: