Sub-cofactorial automorphism-invariant subgroup: Difference between revisions

Latest revision as of 00:31, 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

Property	Meaning	Intermediate notions
Subgroup-cofactorial automorphism-invariant subgroup	invariant under all subgroup-cofactorial automorphisms	\|FULL LIST, MORE INFO
left-transitively 2-subnormal subgroup	whenever the group is 2-subnormal in a bigger group, so is the subgroup	\|FULL LIST, MORE INFO
2-subnormal subgroup	normal subgroup of a normal subgroup	\|FULL LIST, MORE INFO
subnormal subgroup	there is a subnormal series from it to the whole group	\|FULL LIST, MORE INFO

@@ 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===
-* [[Stronger than::Subgroup-cofactorial automorphism-invariant subgroup]]: Also related:
-** [[Stronger than::Left-transitively 2-subnormal subgroup]]
+{| class="sortable" border="1"
-** [[Stronger than::2-subnormal subgroup]]
+! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
-** [[Stronger than::Subnormal subgroup]]
+|-
+| [[Stronger than::Subgroup-cofactorial automorphism-invariant subgroup]] || invariant under all [[subgroup-cofactorial automorphism]]s || || || {{intermediate notions short|subgroup-cofactorial automorphism-invariant subgroup|sub-cofactorial automorphism-invariant subgroup}}
+|-
+| [[Stronger than::left-transitively 2-subnormal subgroup]] || whenever the group is [[2-subnormal subgroup|2-subnormal]] in a bigger group, so is the subgroup || || || {{intermediate notions short|left-transitively 2-subnormal subgroup|sub-cofactorial automorphism-invariant subgroup}}
+|-
+| [[Stronger than::2-subnormal subgroup]] || [[normal subgroup]] of a [[normal subgroup]] || || || {{intermediate notions short|2-subnormal subgroup|sub-cofactorial automorphism-invariant subgroup}}
+|-
+| [[Stronger than::subnormal subgroup]] || there is a [[subnormal series]] from it to the whole group || || || {{intermediate notions short|subnormal subgroup|sub-cofactorial automorphism-invariant subgroup}}
+|}