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.