Levi decomposition: Difference between revisions

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In [[mathematics]], in the field of [[group theory]], a [[subgroup]] ''H'' of a given [[group (mathematics)|group]] ''G'' is a '''subnormal subgroup''' of ''G'' if there is a finite chain of subgroups of the group, each one [[normal subgroup | normal]] in the next, beginning at ''H'' and ending at ''G''.
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In notation, <math>H</math> is <math>k</math>-subnormal in <math>G</math> if there are subgroups
 
:<math>H=H_0,H_1,H_2,\ldots, H_k=G</math>
 
of <math>G</math> such that <math>H_i</math> is normal in <math>H_{i+1}</math> for each <math>i</math>.
 
A subnormal subgroup is a subgroup that is <math>k</math>-subnormal for some positive integer <math>k</math>.
Some facts about subnormal subgroups:
* A 1-subnormal subgroup is a proper [[normal subgroup]] (and vice versa).
* A [[finitely generated group]] is [[nilpotent group|nilpotent]] if and only if each of its subgroups is subnormal.
* Every [[quasinormal subgroup]], and, more generally, every [[conjugate permutable subgroup]], of a finite group is subnormal.
* Every [[pronormal subgroup]] that is also subnormal, is, in fact, normal. In particular, a [[Sylow subgroup]] is subnormal if and only if it is normal.
* Every 2-subnormal subgroup is a [[conjugate permutable subgroup]].
 
The property of subnormality is [[transitive relation|transitive]], that is, a subnormal subgroup of a subnormal
subgroup is subnormal. In fact, the relation of subnormality can be defined as the [[transitive closure]] of the relation of normality.
 
==See also==
 
*[[Normal subgroup]]
*[[Characteristic subgroup]]
*[[Normal core]]
*[[Normal closure]]
*[[Ascendant subgroup]]
*[[Descendant subgroup]]
*[[Serial subgroup]]
 
==References==
* {{Citation | last1=Robinson | first1=Derek J.S. | title=A Course in the Theory of Groups | publisher=[[Springer-Verlag]] | location=Berlin, New York | isbn=978-0-387-94461-6 | year=1996}}
 
[[Category:Subgroup properties]]

Latest revision as of 17:43, 3 December 2014

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