# Symmetric subquandle of a group

From Groupprops

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

## Contents

## Definition

Suppose is a group and is a non-empty subset of . We say that is a **1-closed subquandle** of if the following hold:

- is a symmetric subset of , i.e., it contains the identity element and is closed under taking inverses.
- For any , the conjugate is in . Note that by the preceding, this is equivalent to requiring that for any , the conjugate is in . In particular, is a subquandle of the quandle given by the conjugation rack of .

## Relation with other properties

### Stronger properties

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

subgroup | 1-closed subquandle of a group|FULL LIST, MORE INFO | |||

1-closed subquandle of a group | |FULL LIST, MORE INFO |

### Weaker properties

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

symmetric subset | |FULL LIST, MORE INFO |