Modular subgroup

This subgroup property arises from a property of elements in lattices, when applied to the given subgroup as an element in the lattice of subgroups of a given group.

This is a variation of normality|Find other variations of normality | Read a survey article on varying normality

Definition

Symbol-free definition

A subgroup of a group is termed a modular subgroup if it is a modular element in the lattice of subgroups.

Definition with symbols

A subgroup $A$ of a group $G$ is termed a modular subgroup if for any subgroups $B$ and $C$ of $G$ such that $A\le C$:

$⟨A,B\cap C⟩=⟨A,B⟩\cap C$

Relation with other properties

Stronger properties

• Normal subgroup
• Permutable subgroup: For proof of the implication, refer Permutable implies modular and for proof of its strictness (i.e. the reverse implication being false) refer Modular not implies permutable.
• Distributive subgroup

The proof for permutable subgroups (and hence, for normal subgroups) follows from the modular property of groups.

Weaker properties

• Nilpotent quotient-by-core subgroup
• Permodular subgroup

Metaproperties

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

The whole group is clearly a modular subgroup of itself. So is the trivial subgroup.

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

Suppose $H\le K\le G$ such that $H$ is modular in $G$. Then, clearly, $H$ must be a modular element with respect to all choices of subgroups in $G$, and hence, in particular, in $K$.

Thus, $H$ is also modular in $K$.

For full proof, refer: Modularity satisfies intermediate subgroup condition

Join-closedness

YES: This subgroup property is join-closed: an arbitrary (nonempty) join of subgroups with this property, also has this property.
ABOUT THIS PROPERTY: View variations of this property that are join-closed | View variations of this property that are not join-closed
ABOUT JOIN-CLOSEDNESS: View all join-closed subgroup properties (or, strongly join-closed properties) | View all subgroup properties that are not join-closed | Read a survey article on proving join-closedness | Read a survey article on disproving join-closedness

A join of modular subgroups is modular. This follows from a general lattice-theoretic fact that a join of modular elements is modular.

For full proof, refer: Modularity is strongly join-closed