Existentially bound-word subgroup: Difference between revisions

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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 obtained from 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

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 ==Definition==
-Let <math>C</math> be a collection of pairs <math>(w,l)</math> where <math>w</math> is a word and <math>l</math> is a letter used in <math>w</math>. An element <math>g \in G</math> is said to satisfy the pair <math>(w,l)</math> if we can find values for the other letters of <math>w</math>, with <math>l = g</math>, so that the word <math>w</math> simplifies to the identity element.
+An ''existentially quantified word-letter pair'' a pair <math>(w,l)</math> where <math>w</math> is a word and <math>l</math> is a letter used in <math>w</math>. An element <math>g \in G</math> is said to satisfy the pair <math>(w,l)</math> if we can find values for the other letters of <math>w</math>, with <math>l = g</math>, so that the word <math>w</math> simplifies to the identity element. The subgroup corresponding to such a pair is the subgroup generated by all <math>g \in G</math> satisfying the pair.
-Then, the existentially bound-word subgroup corresponding to <math>C</math> is defined as the subgroup generated by all elements satisfying at least one <math>(w,l) \in C</math>.
+An '''existentially bound-word subgroup''' of a group is a subgroup obtained from subgroups corresponding to existentially quantified word-letter pairs.
-A subgroup of a group is termed an '''existentially bound-word subgroup''' if there exists a collection <math>C</math> for which it is the corresponding existentially bound-word subgroup.
 ==Relation with other properties==