Symmetric subquandle of a group

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

Definition

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

1. $S$ is a symmetric subset of $G$, i.e., it contains the identity element and is closed under taking inverses.
2. For any $a,b\in S$, the conjugate $aba^{-1}$ is in $S$. Note that by the preceding, this is equivalent to requiring that for any $a,b\in S$, the conjugate $a^{-1}ba$ is in $S$. In particular, $S$ is a subquandle of the quandle given by the conjugation rack of $G$.

Relation with other properties

Stronger properties

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