Procharacteristic subgroup: Difference between revisions

Latest revision as of 01:48, 25 February 2009

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Definition

Definition with symbols

(Left-action convention): A subgroup $H$ of a group $G$ is termed procharacteristic in $G$ if, for any automorphism $σ$ of $G$ , there exists $g \in ⟨ H, σ (H) ⟩$ such that $g H g^{- 1} = σ (H)$ .
(Right-action convention): A subgroup $H$ of a group $G$ is termed procharacteristic in $G$ if, for any automorphism $σ$ of $G$ , there exists $g \in ⟨ H, H^{σ} ⟩$ such that $H^{g} = H^{σ}$ .

Relation with other properties

Stronger properties

Weaker properties

Facts

Any procharacteristic subgroup of a normal subgroup is pronormal. Further information: Procharacteristic of normal implies pronormal
A subgroup $H$ of a group $G$ is procharacteristic in $G$ if and only if whenever $G$ is normal in some group $K$ , $H$ is pronormal in $K$ . Further information: Left residual of pronormal by normal is procharacteristic

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 ===Definition with symbols===
-A [[subgroup]] <math>H</math> of a [[group]] <math>G</math> is termed '''procharacteristic''' in <math>G</math> if, for any automorphism <math>\sigma</math> of <math>G</math>, there exists <math>g \in \langle H, \sigma(H) \rangle</math> such that <math>gHg^{-1} = \sigma(H)</math>.
+* (''Left-action convention''): A [[subgroup]] <math>H</math> of a [[group]] <math>G</math> is termed '''procharacteristic''' in <math>G</math> if, for any automorphism <math>\sigma</math> of <math>G</math>, there exists <math>g \in \langle H, \sigma(H) \rangle</math> such that <math>gHg^{-1} = \sigma(H)</math>.
+* (''Right-action convention''): A subgroup <math>H</math> of a group <math>G</math> is termed '''procharacteristic''' in <math>G</math> if, for any automorphism <math>\sigma</math> of <math>G</math>, there exists <math>g \in \langle H, H^\sigma\rangle</math> such that <math>H^g = H^\sigma</math>.
 ==Relation with other properties==
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 ===Stronger properties===
+* [[Weaker than::Characteristic subgroup]]
 * [[Weaker than::Intermediately isomorph-conjugate subgroup]]
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 * [[Stronger than::Automorph-conjugate subgroup]]
+* [[Stronger than::Weakly procharacteristic subgroup]]
 * [[Stronger than::Pronormal subgroup]]
+* [[Stronger than::Weakly pronormal subgroup]]
 ==Facts==