Automorph-conjugacy is not finite-intersection-closed
From Groupprops
This article gives the statement, and possibly proof, of a subgroup property (i.e., automorph-conjugate subgroup) not satisfying a subgroup metaproperty (i.e., finite-intersection-closed subgroup property).
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Statement
We can have a group with two automorph-conjugate subgroups
, such that
is not automorph-conjugate in
.
Proof
Example in the symmetric group
Further information: symmetric group:S6
(This example demonstrates the stronger fact that automorph-conjugacy is not finite-conjugate-intersection-closed).
Let be the symmetric group on six letters:
. Let
be the following 2-Sylow subgroups of
:
In other words, is the internal direct product of a 2-Sylow subgroup on
with the 2-Sylow subgroup on
, while
is the internal direct product of the 2-Sylow subgroup on
with a 2-Sylow subgroup on
.
The intersection is given by:
.
Now, note that:
- Both
and
are automorph-conjugate, because they are both Sylow subgroups, and Sylow implies automorph-conjugate.
-
is not automorph-conjugate. To see this, note that
has an outer automorphism that sends transpositions to triple transpositions. Under this automorphism,
goes to a subgroup of
that contains three commuting triple transpositions. If this is conjugate to
, then
should also contain three commuting triple transpositions. But it doesn't.