Left-transitively 2-subnormal subgroup: Difference between revisions

Latest revision as of 21:49, 30 May 2020

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Definition

A subgroup $H$ of a group $K$ is termed a left-transitively 2-subnormal subgroup if it satisfies the following equivalent conditions:

Whenever $K$ is a 2-subnormal subgroup of a group $G$ , $H$ is also a 2-subnormal subgroup of $G$ .
Whenever $K$ is a normal subgroup of characteristic subgroup of a group $G$ , $H$ is also a normal subgroup of characteristic subgroup of $G$ .
For any automorphism $\sigma$ of $K$ , and any element $g\in H$ , the automorphism of $K$ given as $\sigma \circ c_{g}\circ \sigma ^{-1}$ , where $c_{g}$ denotes conjugation by $g$ , preserves $H$ .

Equivalence of definitions

For full proof, refer: Equivalence of definitions of left-transitively 2-subnormal subgroup

Formalisms

In terms of the left transiter

This property is obtained by applying the left transiter to the property: 2-subnormal subgroup
View other properties obtained by applying the left transiter

In terms of the left transiter

This property is obtained by applying the left transiter to the property: normal subgroup of characteristic subgroup
View other properties obtained by applying the left transiter

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	Characteristic implies left-transitively 2-subnormal		\|FULL LIST, MORE INFO
subgroup-cofactorial automorphism-invariant subgroup		subgroup-cofactorial automorphism-invariant implies left-transitively 2-subnormal	Left-transitively 2-subnormal not implies subgroup-cofactorial automorphism-invariant	\|FULL LIST, MORE INFO
cofactorial automorphism-invariant subgroup		cofactorial automorphism-invariant implies left-transitively 2-subnormal		\|FULL LIST, MORE INFO
sub-cofactorial automorphism-invariant subgroup				\|FULL LIST, MORE INFO

Weaker properties

Property	Meaning	Proof of implication	Intermediate notions
normal subgroup of characteristic subgroup	normal subgroup of a characteristic subgroup	left-transitively 2-subnormal implies normal of characteristic	\|FULL LIST, MORE INFO
2-subnormal subgroup			\|FULL LIST, MORE INFO
left-transitively fixed-depth subnormal subgroup	left-transitively $k$ -subnormal for some $k$		\|FULL LIST, MORE INFO

Incomparable properties

Normal subgroup: For full proof, refer: Normal not implies left-transitively 2-subnormal, Left-transitively 2-subnormal not implies normal
Right-transitively 2-subnormal subgroup

Metaproperties

BEWARE! This section of the article uses terminology local to the wiki, possibly without giving a full explanation of the terminology used (though efforts have been made to clarify terminology as much as possible within the particular context)

Here is a summary:

Metaproperty name	Satisfied?	Proof	Difficulty level (0-5)	Statement with symbols
transitive subgroup property	Yes	left-transitive 2-subnormality is transitive		If $H\leq K\leq G$ are groups such that $H$ is left-transitively 2-subnormal in $K$ and $K$ is left-transitively 2-subnormal in $G$ , then $H$ is left-transitively 2-subnormal in $G$ .
trim subgroup property	Yes	Obvious reasons	0	For any group $G$ , $\{e\}$ and $G$ are characteristic in $G$
strongly intersection-closed subgroup property	Yes	left-transitive 2-subnormality is strongly intersection-closed		If $H_{i},i\in I$ , are all left-transitively 2-subnormal in $G$ , so is the intersection of subgroups $\bigcap _{i\in I}H_{i}$ .
intermediate subgroup condition	No	left-transitive 2-subnormality does not satisfy intermediate subgroup condition		It is possible to have groups $H\leq K\leq G$ such that $H$ is left-transitively 2-subnormal in $G$ but not in $K$ .

@@ Line 4: / Line 4: @@
 ==Definition==
-A [[subgroup]] <math>H</math> of a [[group]] <math>K</math> is termed a '''left-transitively 2-subnormal subgroup''' if whenever <math>K</math> is a [[2-subnormal subgroup]] of a group <math>G</math>, <math>H</math> is also a 2-subnormal subgroup of <math>G</math>.
+A [[subgroup]] <math>H</math> of a [[group]] <math>K</math> is termed a '''left-transitively 2-subnormal subgroup''' if it satisfies the following equivalent conditions:
+# Whenever <math>K</math> is a [[2-subnormal subgroup]] of a group <math>G</math>, <math>H</math> is also a 2-subnormal subgroup of <math>G</math>.
+# Whenever <math>K</math> is a [[normal subgroup of characteristic subgroup]] of a group <math>G</math>, <math>H</math> is also a normal subgroup of characteristic subgroup of <math>G</math>.
+# For any automorphism <math>\sigma</math> of <math>K</math>, and any element <math>g \in H</math>, the automorphism of <math>K</math> given as <math>\sigma \circ c_g \circ \sigma^{-1}</math>, where <math>c_g</math> denotes conjugation by <math>g</math>, preserves <math>H</math>.
+===Equivalence of definitions===
+{{proofat|[[Equivalence of definitions of left-transitively 2-subnormal subgroup]]}}
 ==Formalisms==
 {{obtainedbyapplyingthe|left transiter|2-subnormal subgroup}}
+{{obtainedbyapplyingthe|left transiter|normal subgroup of characteristic subgroup}}
 ==Relation with other properties==
@@ Line 14: / Line 24: @@
 ===Stronger properties===
-* [[Weaker than::Characteristic subgroup]]: {{proofat|[[Characteristic implies left-transitively 2-subnormal]]}}
+{| class="sortable" border="1"
-* [[Weaker than::Cofactorial automorphism-invariant subgroup]]: {{proofat|[[Cofactorial automorphism-invariant implies left-transitively 2-subnormal]]}}
+! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
-* [[Left-transitively normal subgroup of characteristic subgroup]]
+|-
+| [[Weaker than::characteristic subgroup]] || invariant under all [[automorphism]]s ||[[Characteristic implies left-transitively 2-subnormal]] || || {{intermediate notions short|left-transitively 2-subnormal subgroup|characteristic subgroup}}
+|-
+| [[Weaker than::subgroup-cofactorial automorphism-invariant subgroup]]|| ||[[subgroup-cofactorial automorphism-invariant implies left-transitively 2-subnormal]]||[[Left-transitively 2-subnormal not implies subgroup-cofactorial automorphism-invariant]]|| {{intermediate notions short|left-transitively 2-subnormal subgroup|subgroup-cofactorial automorphism-invariant subgroup}}
+|-
+| [[Weaker than::cofactorial automorphism-invariant subgroup]] || || [[cofactorial automorphism-invariant implies left-transitively 2-subnormal]] || || {{intermediate notions short|left-transitively 2-subnormal subgroup|cofactorial automorphism-invariant subgroup}}
+|-
+| [[Weaker than::sub-cofactorial automorphism-invariant subgroup]] || || || || {{intermediate notions short|left-transitively 2-subnormal subgroup|sub-cofactorial automorphism-invariant subgroup}}
+|}
 ===Weaker properties===
-* [[Stronger than::Normal subgroup of characteristic subgroup]]: {{proofat|[[Left-transitively 2-subnormal implies normal subgroup of characteristic 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::normal subgroup of characteristic subgroup]] || [[normal subgroup]] of a [[characteristic subgroup]] || [[left-transitively 2-subnormal implies normal of characteristic]] || || {{intermediate notions short|normal subgroup of characteristic subgroup|left-transitively 2-subnormal subgroup}}
+|-
+| [[Stronger than::2-subnormal subgroup]] || || || || {{intermediate notions short|2-subnormal subgroup|left-transitively 2-subnormal subgroup}}
+|-
+| [[Stronger than::left-transitively fixed-depth subnormal subgroup]]|| left-transitively <math>k</math>-subnormal for some <math>k</math> || || || {{intermediate notions short|left-transitively fixed-depth subnormal subgroup|left-transitively 2-subnormal subgroup}}
+|}
 ===Incomparable properties===
-* [[Normal subgroup]]: {{proofat|[[Normal not implies left-transitively 2-subnormal]]|[[Left-transitively 2-subnormal not implies normal]]}}
+* [[Normal subgroup]]: {{proofat|[[Normal not implies left-transitively 2-subnormal]], [[Left-transitively 2-subnormal not implies normal]]}}
 * [[Right-transitively 2-subnormal subgroup]]
 ==Metaproperties==
-{{transitive}}
+{{wikilocal-section}}
-{{intersection-closed}}
-{{trim}}
-{{join-closed}}
+Here is a summary:
-{{not intsubcondn}}
+{| class="sortable" border="1"
+!Metaproperty name !! Satisfied? !! Proof !! Difficulty level (0-5) !! Statement with symbols
+|-
+|[[satisfies metaproperty::transitive subgroup property]] || Yes || [[left-transitive 2-subnormality is transitive]] || {{#show: left-transitive 2-subnormality is transitive| ?Difficulty level}} || If <math>H\le K \le G</math> are groups such that <math>H</math> is left-transitively 2-subnormal in <math>K</math> and <math>K</math> is left-transitively 2-subnormal in <math>G</math>, then <math>H</math> is left-transitively 2-subnormal in <math>G</math>.
+|-
+| [[satisfies metaproperty::trim subgroup property]] || Yes || Obvious reasons || 0 || For any group <math>G</math>, <math>\{ e \}</math> and <math>G</math> are characteristic in <math>G</math>
+|-
+|[[satisfies metaproperty::strongly intersection-closed subgroup property]] || Yes || [[left-transitive 2-subnormality is strongly intersection-closed]] || {{#show: characteristicity is strongly intersection-closed| ?Difficulty level}}|| If <math>H_i, i \in I</math>, are all left-transitively 2-subnormal in <math>G</math>, so is the [[intersection of subgroups]] <math>\bigcap_{i \in I} H_i</math>.
+|-
+| [[dissatisfies metaproperty::intermediate subgroup condition]] || No || [[left-transitive 2-subnormality does not satisfy intermediate subgroup condition]] || || It is possible to have groups <math>H \le K \le G</math> such that <math>H</math> is left-transitively 2-subnormal in <math>G</math> but not in <math>K</math>.
+|}