# Marginal subgroup of finite type

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

A subgroup of a group is termed a **marginal subgroup of finite type** if it can be described in the following equivalent ways:

- It is the marginal subgroup corresponding to a
*single*word. - It is the marginal subgroup corresponding to a
*finite*collection of words.

## Examples

- The center, and all members of the upper central series, are examples.

## Relation with other properties

### Stronger properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

marginal subgroup of finite index | |FULL LIST, MORE INFO |

### Weaker properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

purely definable subgroup | Purely positively definable subgroup|FULL LIST, MORE INFO | |||

elementarily characteristic subgroup | Purely definable subgroup|FULL LIST, MORE INFO | |||

marginal subgroup | |FULL LIST, MORE INFO | |||

strictly characteristic subgroup | invariant under all surjective endomorphisms | Marginal subgroup, Purely positively definable subgroup|FULL LIST, MORE INFO | ||

characteristic subgroup | invariant under all automorphisms | Elementarily characteristic subgroup, Marginal subgroup, Purely definable subgroup, Purely positively definable subgroup|FULL LIST, MORE INFO | ||

normal subgroup | invariant under all inner automorphisms | Purely definable subgroup, Purely positively definable subgroup|FULL LIST, MORE INFO |