Weakly procharacteristic subgroup: Difference between revisions

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Definition

Definition with symbols

A subgroup $H$ of a group $G$ is termed weakly procharacteristic if for any automorphism $σ$ of $G$ , the following holds: if $K$ denotes the closure of $H$ under the action of the cyclic group generated by $σ$ , there exists $g \in K$ such that $σ (H) = g H g^{- 1}$ .

Relation with other properties

Stronger properties

Weaker properties

Effect of property operators

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The intermediately operator

Applying the intermediately operator to this property gives: intermediately automorph-conjugate subgroup

If $H$ is a subgroup of $G$ that is weakly procharacteristic in every intermediate subgroup of $G$ containing it, then $H$ is an intermediately automorph-conjugate subgroup of $G$ . Conversely, if $H$ is intermediately automorph-conjugate in $G$ , then $H$ is weakly procharacteristic in every intermediate subgroup.

Facts

Any weakly procharacteristic subgroup of a normal subgroup is weakly pronormal. Further information: Weakly procharacteristic of normal implies weakly pronormal
Weak procharacteristicity is the most general property for which this is true. Further information: Left residual of weakly pronormal by normal is weakly procharacteristic

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 * [[Stronger than::Weakly pronormal subgroup]]
+==Effect of property operators==
+{{wikilocal-section}}
+{{applyingoperatorgives|intermediately operator|intermediately automorph-conjugate subgroup}}
+If <math>H</math> is a subgroup of <math>G</math> that is weakly procharacteristic in every intermediate subgroup of <math>G</math> containing it, then <math>H</math> is an intermediately automorph-conjugate subgroup of <math>G</math>. Conversely, if <math>H</math> is intermediately automorph-conjugate in <math>G</math>, then <math>H</math> is weakly procharacteristic in every intermediate subgroup.
 ==Facts==
 * Any weakly procharacteristic subgroup of a normal subgroup is weakly pronormal. {{further|[[Weakly procharacteristic of normal implies weakly pronormal]]}}
 * Weak procharacteristicity is the most general property for which this is true. {{further|[[Left residual of weakly pronormal by normal is weakly procharacteristic]]}}