# 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]
This subgroup property is the version, in a pure group, of the following subgroup property for logicians: definable subgroup | See other such examples
This is a variation of characteristicity|Find other variations of characteristicity | Read a survey article on varying characteristicity

## 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.

## Metaproperties

Metaproperty name Satisfied? Proof Statement with symbols
transitive subgroup property Yes pure definability is transitive 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 $G$.
trim subgroup property Yes (obvious) The trivial subgroup and whole group are purely definable.
strongly finite-intersection-closed subgroup property Yes pure definability is strongly finite-intersection-closed Suppose $G$ is a group and $H,K$ are purely definable subgroups of $G$. Then the intersection $H \cap K$ is also a purely definable subgroup of $G$.
quotient-transitive subgroup property Yes pure definability is quotient-transitive Suppose $G$ is a group and $H \le K \le G$ are such that $H$ is purely definable in $G$ and the quotient group $K/H$ is purely definable in $G/H$. Then, $K$ is purely definable in $G$.

## Relation with other properties

### Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
characteristic subgroup of finite group characteristic subgroup and the whole group is a finite group |FULL LIST, MORE INFO
verbal subgroup of finite type image of a word map Purely positively definable subgroup|FULL LIST, MORE INFO
finite verbal subgroup (via verbal subgroup of finite type) (via verbal subgroup of finite type) Verbal subgroup of finite type|FULL LIST, MORE INFO
marginal subgroup of finite type Purely positively definable subgroup|FULL LIST, MORE INFO

### Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
purely definably generated subgroup generated by a purely definable subset derived subgroup not is purely definable, but it is purely definably generated |FULL LIST, MORE INFO
elementarily characteristic subgroup no other elementarily equivalently embedded subgroup Purely definably generated subgroup|FULL LIST, MORE INFO
second-order purely definable subgroup definable in the second-order theory of the group |FULL LIST, MORE INFO
characteristic subgroup invariant under all automorphisms Elementarily characteristic subgroup, Monadic second-order characteristic subgroup, Purely definably generated subgroup|FULL LIST, MORE INFO
normal subgroup invariant under all inner automorphisms Characteristic subgroup|FULL LIST, MORE INFO