# Weakly marginal subgroup

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

### For a single word-letter pair

Suppose $w$ is a word in letters $x_1,x_2,\dots,x_n$ and $x_1$ is a letter of $w$ (we can take the letter to be $x_1$ without loss of generality). The weakly marginal subgroup of a group $G$ corresponding to $w$ and $x_1$ is the subgroup: $\{ g \in G \mid w(gx_1,x_2,\dots,x_n) = w(x_1,x_2,\dots,x_n) = w(x_1g,x_2,\dots,x_n) \ \forall \ x_1,x_2,\dots,x_n \in G \}$

### For a collection of many words and chosen letters in each

The weakly marginal subgroup corresponding to a collection of word-letter pairs is defined as the intersection of the weakly marginal subgroups corresponding to each word-letter pair.

## Examples

Most of the examples are common with marginal subgroup#Examples. In addition, we have some other examples, such as centralizer of derived subgroup, which is weakly marginal but not necessarily marginal.

## Metaproperties

Metaproperty name Satisfied? Proof Statement with symbols
quotient-transitive subgroup property Yes weak marginality is quotient-transitive If $H \le K \le G$ are groups such that $H$ is weakly marginal in $G$ and $K/H$ is weakly marginal in $G/H$, then $K$ is weakly marginal in $G$.
strongly intersection-closed subgroup property Yes weak marginality is strongly intersection-closed If $H_i, i \in I$, are all weakly marginal subgroups of $G$, so is the intersection $\bigcap_{i \in I} H_i$.
transitive subgroup property No weak marginality is not transitive It is possible to have groups $H \le K \le G$ such that $H$ is weakly marginal in $K$ and $K$ is weakly marginal in $G$ but $H$ is not weakly marginal in $G$.
direct power-closed subgroup property Yes weak marginality is direct power-closed Suppose $H$ is a weakly 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 weakly marginal subgroup. In fact, it is weakly marginal for the same collection of word-letter pairs for which $H$ is weakly marginal in $G$.

## Relation with other properties

### Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
marginal subgroup here, we apply the condition to every letter within each word, not just a single letter. marginal implies weakly marginal weakly marginal not implies marginal |FULL LIST, MORE INFO

### Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
bound-word subgroup
strictly characteristic subgroup invariant under all surjective endomorphisms
finite direct power-closed characteristic subgroup in any finite direct power of the group, the corresponding direct power of the subgroup is characteristic
characteristic subgroup invariant under all automorphisms