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 normal 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 equivalent 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