Polynormal subgroup: Difference between revisions

Latest revision as of 22:52, 22 November 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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VIEW FACTS THAT:use, or start with, a subgroup satisfying the property|prove, or show that, a subgroup satisfies the property

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 $g\in G$ , $H$ is a contranormal subgroup in the subgroup $H^{\langle g\rangle }$ , i.e., the closure of $H$ under the action by conjugation of the cyclic subgroup generated by $g$ .

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 $K$ . For full proof, refer: Polynormality satisfies intermediate subgroup condition

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.

Join-closedness

YES: This subgroup property is join-closed: an arbitrary (nonempty) join of subgroups with this property, also has this property.
ABOUT THIS PROPERTY: View variations of this property that are join-closed | View variations of this property that are not join-closed
ABOUT JOIN-CLOSEDNESS: View all join-closed subgroup properties (or, strongly join-closed properties) | View all subgroup properties that are not join-closed | Read a survey article on proving join-closedness | Read a survey article on disproving join-closedness

In fact, an arbitrary, possibly empty, join of polynormal subgroups is polynormal. For full proof, refer: Polynormality is strongly join-closed

Testing

GAP code

One can write code to test this subgroup property in GAP (Groups, Algorithms and Programming), though there is no direct command for it.
View the GAP code for testing this subgroup property at: IsPolynormal
View other GAP-codable subgroup properties | View subgroup properties with in-built commands

GAP-codable subgroup property

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

@@ Line 7: / Line 7: @@
 ===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^{\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> and <math>g^{-1}</math>.
+A [[subgroup]] <math>H</math> of a [[group]] <math>G</math> is termed '''polynormal''' if given any <math>g \in G</math>, <math>H</math> is a [[contranormal subgroup]] in the subgroup <math>H^{\langle g \rangle}</math>, i.e., the closure of <math>H</math> under the action by conjugation of the cyclic subgroup generated by <math>g</math>.
 ==Relation with other properties==
@@ Line 13: / Line 13: @@
 ===Stronger properties===
-* [[Normal subgroup]]
+* [[Weaker than::Normal subgroup]]
-* [[Maximal subgroup]]
+* [[Weaker than::Maximal subgroup]]
-* [[Abnormal subgroup]]
+* [[Weaker than::Abnormal subgroup]]
-* [[Pronormal subgroup]]
+* [[Weaker than::Pronormal subgroup]]
-* [[Weakly abnormal subgroup]]
+* [[Weaker than::Weakly abnormal subgroup]]
-* [[Weakly pronormal subgroup]]
+* [[Weaker than::Weakly pronormal subgroup]]
-* [[Paranormal subgroup]]
+* [[Weaker than::Strongly paranormal subgroup]]
-* [[Sylow subgroup]] in a [[finite group]]
+* [[Weaker than::Paranormal subgroup]]
+* [[Weaker than::Strongly polynormal subgroup]]
+* [[Weaker than::Sylow subgroup]] in a [[finite group]]
 ===Weaker properties===
-* [[Fan subgroup]]
+* [[Stronger than::Fan subgroup]]
-* [[Intermediately subnormal-to-normal subgroup]]: {{proofat|[[Polynormal implies intermediately subnormal-to-normal]]}}
+* [[Stronger than::Intermediately subnormal-to-normal subgroup]]: {{proofat|[[Polynormal implies intermediately subnormal-to-normal]]}}
 ==Metaproperties==
@@ Line 31: / Line 33: @@
 {{intsubcondn}}
-If <math>H</math> is polynormal in <math>G</math>, <math>H</math> is also polynormal in any intermediate subgroup <math>K</math>.
+If <math>H</math> is polynormal in <math>G</math>, <math>H</math> is also polynormal in any intermediate subgroup <math>K</math>. {{proofat|[[Polynormality satisfies intermediate subgroup condition]]}}
 {{trim}}
 The whole group and the trivial subgroup are polynormal; in fact they are [[normal subgroup|normal]].
+{{join-closed}}
+In fact, an arbitrary, possibly empty, join of polynormal subgroups is polynormal. {{proofat|[[Polynormality is strongly join-closed]]}}
+==Testing==
+{{GAP code for subgroup property|test = IsPolynormal}}
 ==References==