# Existentially bound-word subgroup

From Groupprops

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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]

## Definition

An *existentially quantified word-letter pair* a pair where is a word and is a letter used in . An element is said to satisfy the pair if we can find values for the other letters of , with , so that the word simplifies to the identity element. The subgroup corresponding to such a pair is the subgroup generated by all satisfying the pair.

An **existentially bound-word subgroup** of a group is a subgroup that can be expressed as an arbitrary join of subgroups obtained as finite intersections of subgroups corresponding to existentially quantified word-letter pairs.