Left-transitively fixed-depth subnormal subgroup
BEWARE! This term is nonstandard and is being used locally within the wiki. [SHOW MORE]
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]
This is a variation of subnormality|Find other variations of subnormality |
Definition
A subgroup of a group
is termed left-transitively fixed-depth subnormal in
if there exists a natural number
such that
is left-transitively
-subnormal in
. In other words, whenever
is a
-subnormal subgroup of a group
,
is also
-subnormal in
.
Note that any subgroup that is left-transitively -subnormal is also left-transitively
-subnormal for
.
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; for characteristic subgroups, we can set ![]() |
Characteristic of normal implies normal | Left-transitively 2-subnormal subgroup|FULL LIST, MORE INFO | |
Left-transitively 2-subnormal subgroup | obtained by setting ![]() |
|FULL LIST, MORE INFO | ||
Cofactorial automorphism-invariant subgroup | Left-transitively 2-subnormal subgroup|FULL LIST, MORE INFO | |||
Subgroup-cofactorial automorphism-invariant subgroup | Left-transitively 2-subnormal subgroup|FULL LIST, MORE INFO |
Weaker properties
Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|
Subnormal subgroup | Normal not implies left-transitively fixed-depth subnormal | |FULL LIST, MORE INFO |
Related properties
Property | Meaning | Proof of one non-implication | Proof of other non-implication | Notions stronger than both | Notions weaker than both |
---|---|---|---|---|---|
Right-transitively fixed-depth subnormal subgroup | |FULL LIST, MORE INFO | Subnormal subgroup|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
If are such that
is left-transitively
-subnormal in
and
is left-transitively
-subnormal in
, then
is left-transitively
-subnormal in
.
Intersection-closedness
This subgroup property is finite-intersection-closed; a finite (nonempty) intersection of subgroups with this property, also has this property
View a complete list of finite-intersection-closed subgroup properties
An intersection of finitely many such subgroups again has the property. In particular, the intersection of a left-transitively -subnormal subgroup and a left-transitively
-subnormal subgroup is left-transitively
-subnormal.
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
A join of finitely many such subgroups again has the property. In particular, the join of a left-transitively -subnormal subgroup and a left-transitively
-subnormal subgroup is left-transitively
-subnormal.