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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Definition

Definition with symbols

A quantified word-letter pair is a word with a distinguished letter, with a sequence of (possibly nested) existential and universal quantifiers on all the other letters. An element $g$ in a group $G$ satisfies this pair if setting the distinguished letter equal to $g$, the nested expression is true in the group (where existential and universal quantifiers are interpreted over $G$). The subgroup corresponding to a quantified word-letter pair is defined as the subgroup generated by all elements of the group satisfying that pair.

A bound-word subgroup is a possibly arbitrary join of finite intersections of subgroups corresponding to quantified word-letter pairs.

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
verbal subgroup	generated by a set of words			|FULL LIST, MORE INFO
existentially bound-word subgroup	similar definition, but we use only existential quantifiers			|FULL LIST, MORE INFO
marginal subgroup				|FULL LIST, MORE INFO
weakly marginal subgroup				|FULL LIST, MORE INFO

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
strictly characteristic subgroup		Bound-word implies strictly characteristic		|FULL LIST, MORE INFO
characteristic subgroup		(via strictly characteristic)	(via strictly characteristic)	|FULL LIST, MORE INFO
normal subgroup		(via characteristic)	(via characteristic)	Characteristic subgroup|FULL LIST, MORE INFO