Core-characteristic subgroup: Difference between revisions

Revision as of 23:40, 11 January 2010

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Definition

QUICK PHRASES: intersection of all conjugates is characteristic, normal core is characteristic

Symbol-free definition

A subgroup of a group is termed core-characteristic if its normal core is a characteristic subgroup of the whole group.

Definition with symbols

A subgroup $H$ of a group $G$ is termed core-characteristic if the normal core $H_{G}$ of $H$ in $G$ is a characteristic subgroup of $G$ .

Relation with other properties

Stronger properties

Property	Meaning	Intermediate notions
Characteristic subgroup	invariant under all automorphisms	\|FULL LIST, MORE INFO
Automorph-conjugate subgroup	every automorphic subgroup is conjugate to it	\|FULL LIST, MORE INFO
Intersection of automorph-conjugate subgroups	intersection of automorph-conjugate subgroups	\|FULL LIST, MORE INFO
Core-free subgroup	normal core is trivial	\|FULL LIST, MORE INFO

Conjunction with other properties

Any normal subgroup that is also core-characteristic, is characteristic.

Incomparable properties

Closure-characteristic subgroup

@@ Line 18: / Line 18: @@
 ===Stronger properties===
-* [[Characteristic subgroup]]
+{| class="sortable" border"1"
-* [[Automorph-conjugate subgroup]]
+! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
-* [[Intersection of automorph-conjugate subgroups]]
+|-
-* [[Core-free subgroup]]
+| [[Weaker than::Characteristic subgroup]] || invariant under all [[automorphism]]s|| || || {{intermediate notions short|core-characteristic subgroup|characteristic subgroup}}
+|-
+| [[Weaker than::Automorph-conjugate subgroup]] || every [[automorphic subgroups|automorphic subgroup]] is [[conjugate subgroups|conjugate]] to it || || || {{intermediate notions short|core-characteristic subgroup|automorph-conjugate subgroup}}
+|-
+| [[Weaker than::Intersection of automorph-conjugate subgroups]] || intersection of [[automorph-conjugate subgroup]]s || || || {{intermediate notions short|core-characteristic subgroup|intersection of automorph-conjugate subgroups}}
+|-
+| [[Weaker than::Core-free subgroup]] || [[normal core]] is trivial || || || {{intermediate notions short|core-characteristic subgroup|core-free subgroup}}
+|}
 ===Conjunction with other properties===