Existence-bound-word subgroup

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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]


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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 E be a collection of equations, 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 E. A subgroup obtained in this way is called an existence-bound-word subgroup.

Relation with other properties

Stronger properties

Weaker properties