# Algebraic 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]

WARNING: POTENTIAL TERMINOLOGICAL CONFUSION: Please don't confuse this with algebraic group

## Contents

## Definition

A subgroup of a group is termed an **algebraic subgroup** if it is an algebraic subset of the group, i.e., it is an intersection of finite unions of elementary algebraic subsets.

## Relation with other properties

### Stronger properties

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

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

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

c-closed subgroup | centralizer of a subset. | c-closed implies algebraic | |FULL LIST, MORE INFO | |

universally bound-word subgroup | |FULL LIST, MORE INFO | |||

finite subgroup | |FULL LIST, MORE INFO |

### Weaker properties

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

unconditionally closed subgroup | closed in any T0 topological group structure on the whole group. | |FULL LIST, MORE INFO |