Weakly abnormal subgroup: Difference between revisions

Latest revision as of 18:54, 7 October 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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Definition

Symbol-free definition

A subgroup of a group is termed weakly abnormal or upward-closed self-normalizing or intermediately contranormal if it satisfies the following equivalent conditions:

Every element of the group lies in the closure of this subgroup under the action by conjugation by the cyclic subgroup generated by that element.
Every subgroup containing that subgroup is a self-normalizing subgroup of the whole group.
The subgroup is a contranormal subgroup in every intermediate subgroup.

Definition with symbols

A subgroup $H$ of a group $G$ is termed weakly abnormal or upward-closed self-normalizing or intermediately contranormal if it satisfies the following equivalent conditions:

Given any $g \in G$ , $g \in H^{⟨ g ⟩}$ . Here $H^{⟨ g ⟩}$ is the smallest subgroup of $G$ containing $H$ , which is closed under the action by conjugation by the cyclic subgroup generated by $g$
If $H \leq K \leq G$ , then $K$ is a self-normalizing subgroup of $G$ .
If $H \leq L \leq G$ , then $H$ is a contranormal subgroup of $L$ .

Equivalence of definitions

For full proof, refer: Equivalence of definitions of weakly abnormal subgroup

Formalisms

In terms of the upward-closure operator

This property is obtained by applying the upward-closure operator to the property: self-normalizing subgroup
View other properties obtained by applying the upward-closure operator

$H$ is weakly abnormal in $G$ if and only if every subgroup of $G$ containing $H$ is self-normalizing.

In terms of the intermediately operator

This property is obtained by applying the intermediately operator to the property: contranormal subgroup
View other properties obtained by applying the intermediately operator

$H$ is weakly abnormal in $G$ if and only if $H$ is contranormal in every intermediate subgroup.

Relation with other properties

Stronger properties

Non-normal maximal subgroup
Abnormal subgroup

Weaker properties

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 a subgroup is weakly abnormal in the whole group, it is also weakly abnormal in every intermediate subgroup. For full proof, refer: Weak abnormality satisfies intermediate subgroup condition]]

Upward-closedness

This subgroup property is upward-closed: if a subgroup satisfies the property in the whole group, every intermediate subgroup also satisfies the property in the whole group
View other upward-closed subgroup properties

If a subgroup is weakly abnormal in the whole group, then every subgroup containing it is also weakly abnormal in the whole group. For full proof, refer: Weak abnormality is upward-closed

@@ Line 2: / Line 2: @@
 ==Definition==
+===Symbol-free definition===
+A [[subgroup]] of a [[group]] is termed '''weakly abnormal''' or '''upward-closed self-normalizing''' or '''intermediately contranormal''' if it satisfies the following equivalent conditions:
+# Every element of the group lies in the closure of this subgroup under the action by conjugation by the cyclic subgroup generated by that element.
+# Every subgroup containing that subgroup is a [[self-normalizing subgroup]] of the whole group.
+# The subgroup is a [[contranormal subgroup]] in every intermediate subgroup.
 ===Definition with symbols===
-A [[subgroup]] <math>H</math> of a [[group]] <math>G</math> is termed '''weakly abnormal''' if, given any <math>g \in G</math>, <math>g \in H^{<g>}</math>. Here <math>H^{<g>}</math> is 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 '''weakly abnormal''' or '''upward-closed self-normalizing''' or '''intermediately contranormal''' if it satisfies the following equivalent conditions:
+# Given any <math>g \in G</math>, <math>g \in H^{\langle g \rangle}</math>. Here <math>H^{\langle g \rangle}</math> is the smallest subgroup of <math>G</math> containing <math>H</math>, which is closed under the action by conjugation by the cyclic subgroup generated by <math>g</math>
+# If <math>H \le K \le G</math>, then <math>K</math> is a [[defining ingredient::self-normalizing subgroup]] of <math>G</math>.
+# If <math>H \le L \le G</math>, then <math>H</math> is a [[defining ingredient::contranormal subgroup]] of <math>L</math>.
+===Equivalence of definitions===
+{{proofat|[[Equivalence of definitions of weakly abnormal subgroup]]}}
+==Formalisms==
+{{obtainedbyapplyingthe|upward-closure operator|self-normalizing subgroup}}
+<math>H</math> is weakly abnormal in <math>G</math> if and only if every subgroup of <math>G</math> containing <math>H</math> is self-normalizing.
+{{obtainedbyapplyingthe|intermediately operator|contranormal subgroup}}
+<math>H</math> is weakly abnormal in <math>G</math> if and only if <math>H</math> is contranormal in every intermediate subgroup.
 ==Relation with other properties==
@@ Line 11: / Line 37: @@
 ===Stronger properties===
-* [[Abnormal subgroup]]
+* Non-normal [[maximal subgroup]]
+* [[Weaker than::Abnormal subgroup]]
 ===Weaker properties===
-* [[Weakly pronormal subgroup]]
+* [[Stronger than::Weakly pronormal subgroup]]
-* [[Polynormal subgroup]]
+* [[Stronger than::Self-normalizing subgroup]]
+* [[Stronger than::Contranormal subgroup]]
+* [[Stronger than::Paracharacteristic subgroup]]
+* [[Stronger than::Paranormal subgroup]]
+* [[Stronger than::Polycharacteristic subgroup]]
+* [[Stronger than::Polynormal subgroup]]
 ==Metaproperties==
@@ Line 22: / Line 54: @@
 {{intsubcondn}}
-If a subgroup is weakly abnormal in the whole group, it is also weakly abnormal in every intermediate subgroup.
+If a subgroup is weakly abnormal in the whole group, it is also weakly abnormal in every intermediate subgroup. {{proofat|Weak abnormality satisfies intermediate subgroup condition]]}}
+{{upward-closed}}
+If a subgroup is weakly abnormal in the whole group, then every subgroup containing it is also weakly abnormal in the whole group. {{proofat|[[Weak abnormality is upward-closed]]}}