# 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

### 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 in a group satisfies this pair if setting the distinguished letter equal to , the nested expression is true in the group (where existential and universal quantifiers are interpreted over ). 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 |