Purely definable 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 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
Contents
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 are groups and is purely definable in and is purely definable in , then is purely definable in . In fact, we can actually compose a formula defining within with a formula defining within . |
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 is a group and are purely definable subgroups of . Then the intersection is also a purely definable subgroup of . |
quotient-transitive subgroup property | Yes | pure definability is quotient-transitive | Suppose is a group and are such that is purely definable in and the quotient group is normal in . Then, is purely definable in . |
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 |