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

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 Cofactorial automorphism-invariant subgroup, Sub-cofactorial automorphism-invariant subgroup, Subgroup-cofactorial automorphism-invariant subgroup|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 Sub-cofactorial automorphism-invariant subgroup, Subgroup-cofactorial automorphism-invariant subgroup|FULL LIST, MORE INFO
sub-cofactorial automorphism-invariant subgroup Subgroup-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 subgroup of characteristic subgroup |FULL LIST, MORE INFO
2-subnormal subgroup 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

Incomparable properties

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.