# Symmetric group of degree six or higher is not weak subset-conjugacy-closed in general linear group over rationals

## Contents

## Statement

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.

## Related facts

### Similar facts

- 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.

### Opposite facts

- 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.

## Proof

### Example for

Consider the following two subsets of the symmetric group acting on the set :

and

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.