# 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: $\! w(x_1,x_2,\dots,x_n) = w(x_1,x_2,\dots,gx_i,x_{i+1}, \dots x_n) = w(x_1,x_2,\dots,x_ig,x_{i+1},\dots,x_n) \ \forall \ x_1,x_2,\dots,x_n \in G, \forall i \in \{ 1,2,\dots,n \}$

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 $[x_1,x_2] = x_1x_2x_1^{-1}x_2^{-1}$ (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.
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]

## 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 $K$.
direct power-closed subgroup property Yes marginality is direct power-closed Suppose $H$ is a marginal subgroup of a group $G$, and $\alpha$ is a finite or infinite cardinal. Then, in the direct power $G^\alpha$, $H^\alpha$ is a marginal subgroup. In fact, it is marginal for the same variety that $H$ is marginal in $G$.

## Relation with other properties

### Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
marginal subgroup of finite type we use only finitely many words |FULL LIST, MORE INFO

### 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 (see definition)
algebraic subgroup algebraic subset, i.e., intersection of elementary algebraic subsets |FULL LIST, MORE INFO
unconditionally closed subgroup (via algebraic) (via algebraic) Algebraic subgroup|FULL LIST, MORE INFO
strictly characteristic subgroup invariant under all surjective endomorphisms
characteristic subgroup invariant under all automorphisms Bound-word subgroup, Direct power-closed characteristic subgroup, Finite direct power-closed characteristic subgroup, Strictly characteristic subgroup, Sub-weakly marginal subgroup, Submarginal subgroup, Weakly marginal subgroup|FULL LIST, MORE INFO
direct power-closed characteristic subgroup in any direct power of the group, the corresponding direct power of the subgroup is characteristic Finite direct power-closed characteristic subgroup|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 |FULL LIST, MORE INFO

### 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.