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 |