Symmetric group of degree six or higher is not weak subset-conjugacy-closed in general linear group over rationals
Suppose . Consider the symmetric group of degree as a subgroup of the general linear group of degree over the field of rational numbers, denoted , via the embedding as permutation matrices. Then, is not a weak subset-conjugacy-closed subgroup of . In other words, we can find subsets and of that are conjugate as subsets in but not in .
We can take and to be subgroups.
- Analogue of Brauer's permutation lemma fails over rationals for every non-cyclic finite group: In fact, the statement generates a very large collection of examples, though for our purposes we need only one example.
- Brauer's permutation lemma guarantees that any two subgroups of that are conjugate in have an isomorphism between them that preserves cycle types of elements.
Consider the following two subsets of the symmetric group acting on the set :
These are not conjugate inside , because has two global fixed points while has none. However, they are conjugate inside . We can see this either by finding explicit matrices that perform this conjugation, or by noting that the two representations have the same character, and hence, since the Klein four-group is a rational representation group, they must be conjugate over the rational numbers.
Example for higher
We can use the same example as for , fixing all the elements.