Right-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 K of a group G is termed right-transitively 2-subnormal if whenever H is a 2-subnormal subgroup of K, H is also 2-subnormal in G.

Formalisms

In terms of the right transiter

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

Relation with other properties

Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
base of a wreath product base in a wreath product of groups base of a wreath product implies right-transitively 2-subnormal (some of the other stronger properties, like abelian normal subgroup, provide counterexamples) |FULL LIST, MORE INFO
hereditarily 2-subnormal subgroup every subgroup of it is a 2-subnormal subgroup of the whole group. (direct) any group that is not abelian or class two, as a subgroup of itself |FULL LIST, MORE INFO
central subgroup in the center (via hereditarily 2-subnormal) (via hereditarily 2-subnormal) Dedekind normal subgroup, Hereditarily 2-subnormal subgroup, Subgroup of abelian normal subgroup|FULL LIST, MORE INFO
abelian normal subgroup normal subgroup that is also an abelian group (via hereditarily 2-subnormal) (via hereditarily 2-subnormal) Dedekind normal subgroup, Hereditarily 2-subnormal subgroup, Subgroup of abelian normal subgroup|FULL LIST, MORE INFO
subgroup of abelian normal subgroup subgroup inside an abelian normal subgroup (via hereditarily 2-subnormal) (via hereditarily 2-subnormal) Hereditarily 2-subnormal subgroup|FULL LIST, MORE INFO
transitively normal subgroup any normal subgroup of it is normal in the whole group (direct) base of a wreath product that is nontrivial gives a counterexample |FULL LIST, MORE INFO
direct factor factor in an internal direct product (via transitively normal) (via transitively normal) Base of a wreath product|FULL LIST, MORE INFO
central factor product with centralizer is whole group (via transitively normal) (via transitively normal) Base of a wreath product|FULL LIST, MORE INFO

Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
right-transitively fixed-depth subnormal subgroup there is some natural number k such that any k-subnormal subgroup of the subgroup is also k-subnormal in the whole group. (set k = 2). any subgroup of greater subnormal depth than two inside a finite p-group |FULL LIST, MORE INFO
2-subnormal subgroup normal subgroup of a normal subgroup (direct) follows from there exist subgroups of arbitrarily large subnormal depth |FULL LIST, MORE INFO

Metaproperties

Transitivity

This subgroup property is transitive: a subgroup with this property in a subgroup with this property, also has this property in the whole group.
ABOUT THIS PROPERTY: View variations of this property that are transitive | View variations of this property that are not transitive
ABOUT TRANSITIVITY: View a complete list of transitive subgroup properties|View a complete list of facts related to transitivity of subgroup properties |Read a survey article on proving transitivity

Trimness

This subgroup property is trim -- it is both trivially true (true for the trivial subgroup) and identity-true (true for a group as a subgroup of itself).
View other trim subgroup properties | View other trivially true subgroup properties | View other identity-true subgroup properties

Intermediate subgroup condition

YES: This subgroup property satisfies the intermediate subgroup condition: if a subgroup has the property in the whole group, it has the property in every intermediate subgroup.
ABOUT THIS PROPERTY: View variations of this property satisfying intermediate subgroup condition | View variations of this property not satisfying intermediate subgroup condition
ABOUT INTERMEDIATE SUBROUP CONDITION:View all properties satisfying intermediate subgroup condition | View facts about intermediate subgroup condition