Marginal subgroup

Definition

Marginal subgroup for a single word

Suppose $w$ is a word in the letters $x_1,x_2,\dots ,x_n$ and $G$ is a group. The marginal subgroup for $w$ in $G$ is the set of all $g\in G$ such that:

$w(x_1,x_2,\dots ,x_n)=w(x_1,x_2,\dots ,g{x}_i,x_{i+1},\dots x_n)=w(x_1,x_2,\dots ,x_{i}g,x_{i+1},\dots ,x_n)\mathrm{\forall }{x}_1,x_2,\dots ,x_n\in G$

That this set is a subgroup is readily verified.

Marginal subgroup for a collection of words

The marginal subgroup for a (possibly infinite) collection of words in a group is the intersection of the marginal subgroups for each of the words in that group.

Examples

Extreme examples

• The whole group is the marginal subgroup for the word $w(x)=x$.
• The trivial subgroup is the marginal subgroup for the empty word, i.e., the word that evaluates to the identity element for any element.

Commutator word and center

• The marginal subgroup for the commutator word $[x_1,x_2]=x_1{x}_2{x}_1^{-1}{x}_2^{-1}$ (this is the left-normed commutator, but the same holds for the right-normed commutator) is precisely the center.
• The marginal subgroup for a left-normed iterated commutator such as $[\dots [x_1,x_2],x_3],\dots ,x_n]$ is the $(n-1{)}^{th}$ member of the upper central series. (This is not completely obvious for $n\ge 3$).

Power words

• The marginal subgroup for the power word $x^2$ is the set of central elements of order dividing 2.

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
member of upper central series (finite part)

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
bound-word subgroup	can be expressed in terms of words and equations
strictly characteristic subgroup	invariant under all surjective endomorphisms
characteristic subgroup	invariant under all automorphisms			|FULL LIST, MORE INFO
finite direct power-closed characteristic subgroup	in any finite direct power of the group, the corresponding direct power of the subgroup is characteristic

Dual property

The notion of marginal subgroup is somewhat dual to the notion of verbal subgroup, which is the subgroup generated by all elements realized using the given word.