# Marginal subgroup

From Groupprops

## Definition

### Marginal subgroup for a single word

Suppose is a word in the letters and is a group. The **marginal subgroup** for in is the set of all such that:

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.

## Examples

### Extreme examples

- 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 |

### Power words

- 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]

## Metaproperties

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 . |

## Facts

## 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

### 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.