|
|
| Line 1: |
Line 1: |
| 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''.
| | Hello, my name is Andrew and my wife doesn't like it at all. She is truly fond of caving but she doesn't have the time recently. My working day occupation is an info officer free psychic - [http://www.khuplaza.com/dent/14869889 http://www.khuplaza.com/], but I've already utilized for an additional [http://clothingcarearchworth.com/index.php?document_srl=441551&mid=customer_review psychic readers] 1. Some time in the past she chose to live in Alaska and her mothers and fathers live close by.<br><br>Feel free to visit my web page :: authentic psychic readings ([http://netwk.hannam.ac.kr/xe/data_2/85669 netwk.hannam.ac.kr]) |
| | |
| 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]]
| |
Hello, my name is Andrew and my wife doesn't like it at all. She is truly fond of caving but she doesn't have the time recently. My working day occupation is an info officer free psychic - http://www.khuplaza.com/, but I've already utilized for an additional psychic readers 1. Some time in the past she chose to live in Alaska and her mothers and fathers live close by.
Feel free to visit my web page :: authentic psychic readings (netwk.hannam.ac.kr)