Weakly marginal subgroup

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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:

$\{g\in G\mid w(gx_{1},x_{2},\dots ,x_{n})=w(x_{1},x_{2},\dots ,x_{n})=w(x_{1}g,x_{2},\dots ,x_{n})\ \forall \ x_{1},x_{2},\dots ,x_{n}\in G\}$

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 $H\leq K\leq G$ 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 $H\leq K\leq G$ 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