Center of pronormal subgroup: Difference between revisions

Latest revision as of 23:09, 11 July 2010

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]
|Get subgroup property lookup help |Get exploration suggestions
VIEW RELATED: Subgroup property implications | Subgroup property non-implications | Subgroup metaproperty satisfactions | Subgroup metaproperty dissatisfactions | Subgroup property satisfactions |Subgroup property dissatisfactions
VIEW FACTS THAT:use, or start with, a subgroup satisfying the property|prove, or show that, a subgroup satisfies the property

Definition

A subgroup $H$ of a group $G$ is termed a center of pronormal subgroup if there exists a pronormal subgroup $K$ of $G$ such that $H$ equals the center of $K$ .

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
abelian pronormal subgroup				\|FULL LIST, MORE INFO
center of Sylow subgroup				\|FULL LIST, MORE INFO

Weaker properties

Property	Meaning	Proof of implication	Intermediate notions
characteristic central subgroup of pronormal subgroup	characteristic central subgroup of pronormal subgroup (characteristic being in the pronormal subgroup)
central subgroup of pronormal subgroup
SCDIN-subgroup	subset-conjugacy-determined subgroup in its normalizer	center of pronormal implies SCDIN	\|FULL LIST, MORE INFO

@@ Line 9: / Line 9: @@
 ===Stronger properties===
-* [[Weaker than::Abelian pronormal subgroup]]
+{| class="sortable" border="1"
-* [[Weaker than::Pronormal subgroup]]
+! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
-* [[Weaker than::Center of Sylow subgroup]]
+|-
+| [[Weaker than::abelian pronormal subgroup]] || || || || {{intermediate notions short|center of pronormal subgroup|abelian pronormal subgroup}}
+|-
+| [[Weaker than::center of Sylow subgroup]] || || || || {{intermediate notions short|center of pronormal subgroup|center of Sylow subgroup}}
+|}
 ===Weaker properties===
-* [[Stronger than::Characteristic central subgroup of pronormal subgroup]]
+{| class="sortable" border="1"
-* [[Stronger than::Central subgroup of pronormal subgroup]]
+! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
-* [[Stronger than::SCDIN-subgroup]]: {{proofat|[[Center of pronormal implies SCDIN]]}}
+|-
+| [[Stronger than::characteristic central subgroup of pronormal subgroup]] || [[characteristic subgroup|characteristic]] [[central subgroup]] of [[pronormal subgroup]] (characteristic being ''in'' the pronormal subgroup) || || ||
+|-
+| [[Stronger than::central subgroup of pronormal subgroup]] || || || ||
+|-
+| [[Stronger than::SCDIN-subgroup]] || [[subset-conjugacy-determined subgroup]] in its [[normalizer]] ||[[center of pronormal implies SCDIN]] || || {{intermediate notions short|SCDIN-subgroup|center of pronormal subgroup}}
+|}