Monadic second-order subgroup property

Symbol-free definition
A subgroup property is termed a monadic second-order group property if it can be expressed via monadic second-order logic; in other words, it can be expressed in a language which allows for:


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

Since we can quantify over all subsets, and can test closure under group operations, we can also quantify over all subgroups.

Stronger metaproperties

 * First-order subgroup property

Weaker metaproperties

 * Second-order subgroup property