# Weakly marginal subgroup

From Groupprops

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]

## Contents

## Definition

### For a single word-letter pair

Suppose is a word in letters and is a letter of (we can take the letter to be without loss of generality). The **weakly marginal subgroup** of a group corresponding to and is the subgroup:

### 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 are groups such that is weakly marginal in and is weakly marginal in , then is weakly marginal in . |

strongly intersection-closed subgroup property | Yes | weak marginality is strongly intersection-closed | If , are all weakly marginal subgroups of , so is the intersection . |

transitive subgroup property | No | weak marginality is not transitive | It is possible to have groups such that is weakly marginal in and is weakly marginal in but is not weakly marginal in . |

direct power-closed subgroup property | Yes | weak marginality is direct power-closed | Suppose is a weakly marginal subgroup of a group , and is a finite or infinite cardinal. Then, in the direct power , is a weakly marginal subgroup. In fact, it is weakly marginal for the same collection of word-letter pairs for which is weakly marginal in . |

## 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 |