Weakly marginal subgroup

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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
[[Stronger than::finite direct power-closed characteristic subgroup in any finite direct power of the group, the corresponding direct power of the subgroup is characteristic.
Stronger than::characteristic subgroup invariant under all automorphisms