Subgroup contained in finitely many intermediate subgroups
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]
Contents
Definition
Symbol-free definition
A subgroup of a group is said to be contained in finitely many intermediate subgroups if the number of subgroups of the whole group containing that subgroup is finite.
Definition with symbols
A subgroup of a group
is said to be contained in finitely many intermediate subgroups if the number of subgroups
of
such that
is finite.
Relation with other properties
Stronger properties
Facts
- If a normal subgroup is contained in only finitely many intermediate subgroups, it has finite index. This follows from the fourth isomorphism theorem and the fact that a group having finitely many subgroups is finite.
Metaproperties
Transitivity
NO: This subgroup property is not transitive: a subgroup with this property in a subgroup with this property, need not have the 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 subgroup properties that are not transitive|View facts related to transitivity of subgroup properties | View a survey article on disproving transitivity
If are groups such that
is contained in only finitely many intermediate subgroups of
and
is contained in only finitely many intermediate subgroups of
, it may well happen that
is contained in infinitely many intermediate subgroups of
.
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
If are groups such that
is contained in only finitely many intermediate subgroups of
, then
is also contained in only finitely many intermediate subgroups of
.
Upward-closedness
This subgroup property is upward-closed: if a subgroup satisfies the property in the whole group, every intermediate subgroup also satisfies the property in the whole group
View other upward-closed subgroup properties
If are groups such that
is contained in only finitely many intermediate subgroups of
, then
is also contained in only finitely many intermediate subgroups of
.