# Conjugacy-closed not implies weak subset-conjugacy-closed

From Groupprops

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) neednotsatisfy the second subgroup property (i.e., weak subset-conjugacy-closed subgroup)

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## Statement

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 .

## Facts used

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

## Proof

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.