# Modular subgroup

From Groupprops

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 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

## Contents

## 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 of a group is termed a **modular subgroup** if for any subgroups and of such that :

## 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.ABOUT THIS PROPERTY: View variations of this property satisfying intermediate subgroup condition | View variations of this property not satisfying intermediate subgroup conditionABOUT INTERMEDIATE SUBROUP CONDITION:View all properties satisfying intermediate subgroup condition | View facts about intermediate subgroup condition

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

Thus, is also modular in .

`For full proof, refer: Modularity satisfies intermediate subgroup condition`