Existentially bound-word subgroup

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Definition

An existentially quantified word-letter pair a pair $(w,l)$ where $w$ is a word and $l$ is a letter used in $w$. An element $g\in G$ is said to satisfy the pair $(w,l)$ if we can find values for the other letters of $w$, with $l=g$, so that the word $w$ simplifies to the identity element. The subgroup corresponding to such a pair is the subgroup generated by all $g\in G$ 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.

Relation with other properties

Stronger properties

• Verbal subgroup

Weaker properties

• Bound-word subgroup
• Fully characteristic subgroup: For full proof, refer: Existentially bound-word implies fully characteristic
• Strictly characteristic subgroup