Modular subgroup

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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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

A subgroup of a group is termed a modular subgroup if it is a modular element in the lattice of subgroups. Explicitly, a subgroup $A$ of a group $G$ is termed a modular subgroup if for any subgroups $B$ and $C$ of $G$ such that $A \le C$:

$\langle A, B \cap C \rangle = \langle A,B \rangle \cap C$

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
permutable subgroup	permutes (commutes) with every subgroup, i.e., its product with any subgroup is a subgroup	permutable implies modular (proof relies on the modular property of groups)	modular not implies permutable	|FULL LIST, MORE INFO
normal subgroup	permutes (commutes) with every element, i.e., its left cosets are the same as its right cosets	(via permutable)	(via permutable)	Modular 2-subnormal subgroup, Modular subnormal subgroup, Permutable subgroup|FULL LIST, MORE INFO
maximal subgroup	proper subgroup not contained in any bigger proper subgroup	maximal implies modular	obvious from the fact that there are normal subgroups that are not maximal, and normal implies modular	|FULL LIST, MORE INFO

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
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.
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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