# Left-transitively 2-subnormal subgroup

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This article defines a subgroup property: a property that can be evaluated to true/false given a group and a subgroup thereof, invariant under subgroup equivalence. View a complete list of subgroup properties[SHOW MORE]

## 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

### 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 subgroup of characteristic subgroup |FULL LIST, MORE INFO
left-transitively fixed-depth subnormal subgroup left-transitively $k$-subnormal for some $k$ |FULL LIST, MORE INFO

## 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 $H\le K \le G$ 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 1 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 $H \le K \le G$ such that $H$ is left-transitively 2-subnormal in $G$ but not in $K$.