- 1 Definition
- 2 Examples
- 3 Metaproperties
- 4 Facts
- 5 Relation with other properties
Marginal subgroup for a single word
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:
- It is the marginal subgroup for the collection of all words that become trivial in that variety.
- 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.
- The whole group is the marginal subgroup for the word .
- 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 (this is the left-normed commutator, but the same holds for the right-normed commutator)||abelian groups|
|second center||groups of nilpotency class at most two|
|finite member of upper central series||nilpotent groups of class at most|
|marginal subgroup for variety of metabelian groups||metabelian groups, i.e., groups of derived length at most two|
- The marginal subgroup for the power word 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]
|Metaproperty name||Satisfied?||Proof||Statement with symbols|
|strongly intersection-closed subgroup property||Yes||marginality is strongly intersection-closed||If are all marginal subgroups of , then so is .|
|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 such that is marginal in and is marginal in but is not marginal in .|
|intermediate subgroup condition||No||marginality does not satisfy intermediate subgroup condition||It is possible to have groups such that is marginal in but not in .|
|direct power-closed subgroup property||Yes||marginality is direct power-closed||Suppose is a marginal subgroup of a group , and is a finite or infinite cardinal. Then, in the direct power , is a marginal subgroup. In fact, it is marginal for the same variety that is marginal in .|
Relation with other 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|
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.