Left-transitively 2-subnormal subgroup

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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:

1. Whenever $K$ is a 2-subnormal subgroup of a group $G$, $H$ is also a 2-subnormal subgroup of $G$.
2. Whenever $K$ is a normal subgroup of characteristic subgroup of a group $G$, $H$ is also a normal subgroup of characteristic subgroup of $G$.
3. 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	Proof of strictness (reverse implication failure)	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 $K$.