Conjugacy-closed not implies weak subset-conjugacy-closed

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This article gives the statement and possibly, proof, of a non-implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property (i.e., conjugacy-closed subgroup) need not satisfy the second subgroup property (i.e., weak subset-conjugacy-closed subgroup)
View a complete list of subgroup property non-implications | View a complete list of subgroup property implications
Get more facts about conjugacy-closed subgroup|Get more facts about weak subset-conjugacy-closed subgroup
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Statement

It is possible to have a group G and a subgroup H of G such that H is a conjugacy-closed subgroup of G but not a weak subset-conjugacy-closed subgroup of G. In particular, H is not a Subset-conjugacy-closed subgroup (?) of G.

Facts used

  1. Brauer's permutation lemma
  2. Symmetric group of degree six or higher is not weak subset-conjugacy-closed in general linear group over rationals

Proof

Consider G = GL(6,\mathbb{Q}) and H as the symmetric group of degree six, embedded in G via permutation matrices. By fact (1), H is conjugacy-closed in G, and by fact (2), H is not subset-conjugacy-closed in G.

Analogous examples can be constructed by replacing \mathbb{Q} with a finite field whose characteristic is a large enough prime.