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]

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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 $H$ of a group $G$ is said to be contained in finitely many intermediate subgroups if the number of subgroups $K$ of $G$ such that $H \le K \le G$ is finite.

Relation with other properties

Stronger properties

• Subgroup of finite index
• Subgroup of finite double coset index

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.