|
|
| Line 1: |
Line 1: |
| In [[mathematics]], in the field of [[group theory]], a '''quasinormal subgroup''', or '''permutable subgroup''', is a [[subgroup]] of a [[group (mathematics)|group]] that commutes (permutes) with every other subgroup. The term ''quasinormal subgroup'' was introduced by [[Øystein Ore]] in 1937.
| | Andrew Berryhill is what his wife loves to call him and he totally digs that title. To perform lacross is some thing he would never give up. She works as a travel agent but soon she'll be on her own. Her family members life in Alaska but her spouse desires them to transfer.<br><br>Also visit my web blog - [http://cpacs.org/index.php?document_srl=90091&mid=board_zTGg26 cheap psychic readings] |
| | |
| Two subgroups are said to permute (or commute) if any element from the first
| |
| subgroup, times an element of the second subgroup, can be written as an element of the second
| |
| subgroup, times an element of the first subgroup. That is, <math>H</math> and <math>K</math>
| |
| as subgroups of <math>G</math> are said to commute if ''HK'' = ''KH'', that is, any element of the form <math>hk</math>
| |
| with <math>h \in H</math> and <math>k \in K</math> can be written in the form <math>k'h'</math>
| |
| where <math>k' \in K</math> and <math>h' \in H</math>.
| |
| | |
| Every quasinormal subgroup is a [[modular subgroup]], that is, a modular element in the [[lattice of subgroups]]. This follows from the [[modular property of groups]].
| |
| | |
| A [[conjugate permutable subgroup]] is one that commutes with all its conjugate subgroups. Every quasinormal subgroup is conjugate permutable.
| |
| | |
| Every [[normal subgroup]] is quasinormal, because, in fact, a normal subgroup commutes
| |
| with every element of the group. The converse is not true. For instance, any extension of a cyclic group of prime power order by another cyclic group of prime power order for the same prime, has the property that all its subgroups are quasinormal. However, not all of its subgroups need be normal.
| |
| | |
| Also, every quasinormal | |
| subgroup of a [[finite group]] is a [[subnormal subgroup]]. This follows from the somewhat
| |
| stronger statement that every conjugate permutable subgroup is subnormal, which in turn
| |
| follows from the statement that every maximal conjugate permutable subgroup is normal. (The finiteness
| |
| is used crucially in the proofs.)
| |
| | |
| In any group, every quasinormal subgroup is [[ascendant subgroup|ascendant]].
| |
| | |
| ==External links==
| |
| * [http://www.maths.tcd.ie/pub/ims/bull56/GiG5612.pdf Old, Recent and New Results on Quasinormal subgroups]
| |
| * [http://sciences.aum.edu/~tfoguel/cp.pdf The proof that conjugate permutable subgroups are subnormal]
| |
| | |
| [[Category:Subgroup properties]]
| |
Andrew Berryhill is what his wife loves to call him and he totally digs that title. To perform lacross is some thing he would never give up. She works as a travel agent but soon she'll be on her own. Her family members life in Alaska but her spouse desires them to transfer.
Also visit my web blog - cheap psychic readings