1-closed subquandle of a group

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This article defines a property of subsets of groups
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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 1-closed subset of $G$, i.e., the cyclic subgroup generated by any element of $S$ is in $S$.
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

Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
1-closed subset |FULL LIST, MORE INFO
symmetric subquandle of a group |FULL LIST, MORE INFO