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 condition
ABOUT 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