# Existence-bound-word 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]

BEWARE!This term is nonstandard and is being used locally within the wiki. [SHOW MORE]

## History

*This term is local to the wiki. To learn more about why this name was chosen for the term, and how it does not conflict with existing choice of terminology, refer the talk page*

## Definition

### Symbol-free definition

A subgroup of a group is termed an *existence-bound-word subgroup* if there exists a collection of equations, each in one unknown and several parameters, such that as the parameters vary over elements of the group, the set of possible solutions to each equation, together, form a generating set for the subgroup.

### Definition with symbols

Let be a collection of *equation*s, each equation featuring one unknown and a finite number of parameter variables. For each equation, define the *possible solution set* to be the set of possible values of the unknown variable for which the equation has a solution. Consider the subgroup generated by the union of possible solution sets for each equation in . A subgroup obtained in this way is called an *existence-bound-word subgroup*.