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]

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 Existentially bound-word subgroup|FULL LIST, MORE INFO
existentially bound-word subgroup similar definition, but we use only existential quantifiers |FULL LIST, MORE INFO
marginal subgroup Weakly 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) Finite direct power-closed characteristic subgroup, Strictly characteristic subgroup|FULL LIST, MORE INFO
normal subgroup (via characteristic) (via characteristic) Characteristic subgroup, Finite direct power-closed characteristic subgroup, Strictly characteristic subgroup|FULL LIST, MORE INFO