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:

$w(x_1,x_2,\dots ,x_n)=w(x_1,x_2,\dots ,g{x}_i,x_{i+1},\dots x_n)=w(x_1,x_2,\dots ,x_{i}g,x_{i+1},\dots ,x_n)\mathrm{\forall }{x}_1,x_2,\dots ,x_n\in G,\mathrm{\forall }i\in \{1,2,\dots ,n\}$

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.

Marginal subgroup for a variety

The marginal subgroup for a subvariety of the variety of groups is defined in the following equivalent ways:

1. It is the marginal subgroup for the collection of all words that become trivial in that variety.
2. It is the marginal subgroup for any collection of words that generates the variety, in the sense that a group is in the variety iff all those words are trivial in it.

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

Marginal subgroup	Corresponding word or words	Corresponding variety
center	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)	abelian groups
second center	$[[x_1,x_2],x_3]$	groups of nilpotency class at most two
finite $c^{th}$ member of upper central series	$[\dots [x_1,x_2],x_3],\dots ,x_{c+1}]$	nilpotent groups of class at most $c$
marginal subgroup for variety of metabelian groups	$[[x_1,x_2],[x_3,x_4]]$	metabelian groups, i.e., groups of derived length at most two

Power words

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

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Metaproperties

Metaproperty name	Satisfied?	Proof	Statement with symbols
strongly intersection-closed subgroup property	Yes	marginality is strongly intersection-closed	If $H_i,i\in I$ are all marginal subgroups of $G$, then so is ${\bigcap }_{i\in I}{H}_i$.
trim subgroup property	Yes		Every group is marginal in itself. The trivial subgroup is marginal in every group.
transitive subgroup property	No	marginality is not transitive	It is possible to have groups $H\le K\le G$ such that $H$ is marginal in $K$ and $K$ is marginal in $G$ but $H$ is not marginal in $G$.
intermediate subgroup condition	No	marginality does not satisfy intermediate subgroup condition	It is possible to have groups $H\le K\le G$ such that $H$ is marginal in $G$ but not in $K$.

Facts

• Marginal subgroup is closed in T0 quasitopological group

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
weakly marginal subgroup	like marginal subgroup, but we now only require the marginality for a fixed letter of the word rather than for all letters of the word.	direct	weakly marginal not implies marginal	|FULL LIST, MORE INFO
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.