Purely definable 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 is the version, in a pure group, of the following subgroup property for logicians: definable subgroup | See other such examples

Definition

Symbol-free definition

A subgroup of a group is said to be purely definable if it is definable as a subset in the first-order theory of the pure group. By pure group, we mean the set equipped only with the structure of the group operations and with no additional first-order data supplied.

By definable subset, we mean that there is a first-order formula with one free variable such that the set of elements satisfying the formula is precisely that subset.

Relation with other properties

=Weaker properties

• Purely definably generated subgroup
• Elementarily characteristic subgroup
• Second-order purely definabe subgroup
• Characteristic subgroup

Metaproperties

Transitivity

This subgroup property is transitive: a subgroup with this property in a subgroup with this property, also has this property in the whole group.
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The property of being purely definable is transitive. That is, if $H\le K\le G$ are groups and $K$ is purely definable in $G$ and $H$ is purely definable in $K$, then $H$ is purely definable in $G$. In fact, we can actually compose a formula defining $H$ within $K$ with a formula defining $K$ within $G$.

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 property of being purely definable is satisfied by both the trivial subgroup and the whole group. The formulae are: PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]

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Intersection-closedness

YES: This subgroup property is intersection-closed: an arbitrary (nonempty) intersection of subgroups with this property, also has this property.
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An intersection of purely definable subgroups is purely definable. The formula for the intersection is obtained by taking the logical conjunction of the formulae defining the subgroups.

Join-closedness

A join of purely definable subgroups need not be purely definable. The basic problem is that the join may include products of arbitrary length, and we cannot get a formula of finite length that accounts for all these formulae. However, any finite join of purely definable subgroups is always a purely definably generated subgroup.

Intermediate subgroup condition

NO: This subgroup property does not satisfy the intermediate subgroup condition: it is possible to have a subgroup satisfying the property in the whole group but not satisfying the property in some intermediate subgroup.
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A purely definable subgroup need not be purely definable in every intermediate subgroup.