Sub-cofactorial automorphism-invariant subgroup: Difference between revisions

Revision as of 00:26, 28 March 2016

Definition

A subgroup $H$ of a group $G$ is termed a sub-cofactorial automorphism-invariant subgroup if there exists an ascending chain of subgroups:

$H = H_{0} \leq H_{1} \leq \dots \leq H_{n} = G$

such that each $H_{i}$ is a cofactorial automorphism-invariant subgroup of $G$ .

Formalisms

In terms of the subordination operator

This property is obtained by applying the subordination operator to the property: cofactorial automorphism-invariant subgroup
View other properties obtained by applying the subordination operator

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
characteristic subgroup	invariant under all automorphisms			\|FULL LIST, MORE INFO
cofactorial automorphism-invariant subgroup	invariant under all cofactorial automorphisms		cofactorial automorphism-invariance is not transitive	\|FULL LIST, MORE INFO

Weaker properties

Subgroup-cofactorial automorphism-invariant subgroup: Also related:

@@ Line 15: / Line 15: @@
 ===Stronger properties===
-* [[Weaker than::Characteristic subgroup]]
+{| class="sortable" border="1"
-* [[Weaker than::Cofactorial automorphism-invariant subgroup]]
+! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
+|-
+| [[Weaker than::characteristic subgroup]] || invariant under all [[automorphism]]s || || || {{intermediate notions short|sub-cofactorial automorphism-invariant subgroup|characteristic subgroup}}
+|-
+| [[Weaker than::cofactorial automorphism-invariant subgroup]] || invariant under all [[cofactorial automorphism]]s || || [[cofactorial automorphism-invariance is not transitive]] || {{intermediate notions short|sub-cofactorial automorphism-invariant subgroup|cofactorial automorphism-invariant subgroup}}
+|}
 ===Weaker properties===