Symmetric 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 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 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