Polynormal subgroup: Difference between revisions

Revision as of 19:17, 15 February 2008

This article defines a subgroup property: a property that can be evaluated to true/false given a group and a subgroup thereof, invariant under subgroup equivalence. View a complete list of subgroup properties [SHOW MORE]
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This is a variation of normality|Find other variations of normality | Read a survey article on varying normality

Definition

Definition with symbols

A subgroup $H$ of a group $G$ is termed polynormal if given any $g\in G$ , there exists a $x\in H^{\langle g\rangle }$ such that $H^{\langle x\rangle }=H^{\langle g\rangle }$ . Here $H^{\langle g\rangle }$ denotes the smallest subgroup of $G$ containing $H$ , which is closed under conjugation by $g$ .

Here $H^{g}=gHg^{-1}$ denotes the conjugate of $H$ by $g$ and the angled braces denote the subgroup generated (i.e. join of subgroups).

Relation with other properties

Stronger properties

Weaker properties

Fan subgroup
Intermediately subnormal-to-normal subgroup: For full proof, refer: Polynormal implies intermediately subnormal-to-normal

Metaproperties

Intermediate subgroup condition

YES: This subgroup property satisfies the intermediate subgroup condition: if a subgroup has the property in the whole group, it has the property in every intermediate subgroup.
ABOUT THIS PROPERTY: View variations of this property satisfying intermediate subgroup condition | View variations of this property not satisfying intermediate subgroup condition
ABOUT INTERMEDIATE SUBROUP CONDITION:View all properties satisfying intermediate subgroup condition | View facts about intermediate subgroup condition

If $H$ is polynormal in $G$ , $H$ is also polynormal in any intermediate subgroup $K$ .

Trimness

This subgroup property is trim -- it is both trivially true (true for the trivial subgroup) and identity-true (true for a group as a subgroup of itself).
View other trim subgroup properties | View other trivially true subgroup properties | View other identity-true subgroup properties

The whole group and the trivial subgroup are polynormal; in fact they are normal.

References

On the arrangement of intermediate subgroups by M. S. Ba and Z. I. Borevich
On the arrangement of subgroups by Z. I. Borevich, Zap. Nauchn. Semin. tOMI, 94, 5-12 (1979)
On the lattice of subgroups by Z. I. Borevich and O. N. Macedonska, Zap. Nauchn. Semin. LOMI, 103, 13-19, 1980
Testing of subgroups of a finite group for some embedding properties like pronormality by V. I. Mysovskikh

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 ===Definition with symbols===
-A [[subgroup]] <math>H</math> of a [[group]] <math>G</math> is termed '''polynormal''' if given any <math>g \in G</math>, there exists a <math>x \in H^{\langle g \rangle}</math> such that <math>H^{<x>} = H^{<g>}</math>. Here <math>H^{\langle g \rangle}</math> denotes the smallest subgroup of <math>G</math> containing <math>H</math>, which is closed under conjugation by <math>g</math>.
+A [[subgroup]] <math>H</math> of a [[group]] <math>G</math> is termed '''polynormal''' if given any <math>g \in G</math>, there exists a <math>x \in H^{\langle g \rangle}</math> such that <math>H^{\langle x\rangle} = H^{\langle g \rangle}</math>. Here <math>H^{\langle g \rangle}</math> denotes the smallest subgroup of <math>G</math> containing <math>H</math>, which is closed under conjugation by <math>g</math>.
+Here <math>H^g = gHg^{-1}</math> denotes the  conjugate of <math>H</math> by <math>g</math> and the angled braces denote the subgroup generated (i.e. [[join of subgroups]]).
 ==Relation with other properties==