First-order subgroup property

Symbol-free definition
A subgroup property is said to be a first-order subgroup property if it can be expressed using a first-order formula, viz a formula that allows:


 * Logical operations (conjunction, disjunction, negation, and conditionals)
 * Equality testing
 * Quantification over elements of the group and subgroup (this in particular allows one to test membership of an element of the group, in the subgroup)
 * Group operations (multiplication, inversion and the identity element)

Things that are not allowed are quantification over other subgroups, quantification over automorphisms, and quantification over supergroups.

Importance
First-order language is severely constricted, at least when it comes to subgroup properties. Hence, not only are there very few first-order subgroup properties of interest, also, very few of the subgroup property operators preserve the first-order nature.

Normality
Normality is a first-order subgroup property as can be seen from the following definition: a subgroup $$N$$ of a group $$G$$ is termed normal if the following holds:

$$\forall g \in G,h \in N, ghg^{-1} \in N$$

The formula is universal of quantifier rank 1.

Centrality
A subgroup is a central subgroup if it lies inside the center, or equivalently, if every element in the subgroup commutes with every element in the group.

Clearly, the property of being a central subgroup is first-order.

The formula is universal of quantifier rank 1.

Central factor
A subgroup is a central factor if every element in the group can be expressed as a product of an element in the subgroup and an element in the centralizer. This can naturally be expressed as a first-order formula of quantifier rank 3 with the outermost layer being universal.

$$\forall g \in G (\exists h \in H, k \in G (\forall m \in H, km = mk))$$

Function restriction formalism
The general question of interest: given a subgroup property with a function restriction expression $$a \to b$$, can we use the expression to give a first-order definition for the subgroup property? It turns out that the following suffice:


 * $$a$$ should be a first-order enumerable function property (this condition is much stronger than just being a first-order function property because we are not allowed to directly quantify over functions.
 * $$b$$ should be a first-order function property in the sense that given any function, it must be possible to give a first-order formula that outputs whether or not the function satisfies $$b$$.

The primary example of a first-order enumerable function property is the property of being an inner automorphism. Most function properties that we commonly enoucnter are first-order (that is, they can be tested/verified using first-order formulae).