Conjugacy-closed not implies weak subset-conjugacy-closed
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)
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It is possible to have a group and a subgroup of such that is a conjugacy-closed subgroup of but not a weak subset-conjugacy-closed subgroup of . In particular, is not a Subset-conjugacy-closed subgroup (?) of .
- Brauer's permutation lemma
- Symmetric group of degree six or higher is not weak subset-conjugacy-closed in general linear group over rationals
Consider and as the symmetric group of degree six, embedded in via permutation matrices. By fact (1), is conjugacy-closed in , and by fact (2), is not subset-conjugacy-closed in .
Analogous examples can be constructed by replacing with a finite field whose characteristic is a large enough prime.