# Twisted subgroup

This article defines a property of subsets of groups

View other properties of subsets of groups|View properties of subsets of abelian groups|View subgroup properties

This is a variation of subgroup|Find other variations of subgroup |

## Contents

## Definition

### Definition with symbols

A subset of a group is termed a **twisted subgroup** if it satisfies the following two conditions:

- The identity element belongs to
- For every ,
- Given in , the element is in

Note that the second condition is redundant when is a finite subset of . Since twisted subgroups are usually studied in the context of finite groups, the condition is typically omitted from the definition. It is, however, necessary for the definition to behave nicely for infinite groups. The corresponding definition without this condition is better called twisted submonoid.

## Relation with other properties

### Stronger properties

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

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

2-powered twisted subgroup | twisted subgroup within which every element has a unique square root. | |FULL LIST, MORE INFO |

### Weaker properties

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

1-closed subset | nonempty subset, contains the cyclic subgroup generated by any element in it | |FULL LIST, MORE INFO | ||

symmetric subset | nonempty subset, contains the identity and closed under taking inverses | |FULL LIST, MORE INFO |

## Property theory

### Associates

`Further information: associate of twisted subgroup is twisted subgroup`

Let be a twisted subgroup of . Then, for any in , the sets and are equal and form another twisted subgroup. Such a twisted subgroup is termed an **associate** of . The relation of being associate is an equivalence relation and we are interested in studying twisted subgroups upto the equivalence relation of being associates.

### Intersection

See intersection of twisted subgroups is twisted subgroup.

## References

**PLACEHOLDER FOR INFORMATION TO BE FILLED IN**: [SHOW MORE] from Foguel's article

## External links

- Tuval Foguel's list of publications which include publications on twisted subgroups