Fusion system-equivalent not implies isomorphic
Contents
Statement
It is possible to have two fusion system-equivalent finite groups and
that are not isomorphic.
Proof
Idea behind proof
The idea is as follows: the fusion systems induced by the group provide information on the conjugation action on the -subgroups. However, it does not provide clear information regarding which elements of the group are doing the acting. Thus, we need to construct different groups where the actions on each Sylow subgroup are effectively the same but the acting elements are different.
Generic construction
Consider two primes such that
divides
. Let
be the cyclic group of order
. Consider a group
of order
that has more than one equivalence class up to automorphisms of subgroups of order
. Let
be representatives of two distinct equivalence classes of such subgroups.
Now define:
where the action of on
is given by a homomorphism
with kernel
. Note that the image is the unique cyclic subgroup of order
inside
, which is cyclic of order
.
Also define:
where the action of on
is given by a homomorphism
with kernel
. Note that the image is the unique cyclic subgroup of order
inside
, which is cyclic of order
.
We note that:
- Both
and
have order
. Moreover, the
-Sylow subgroups of both are isomorphic to
(cyclic of order
) and the
-Sylow subgroups are isomorphic to
(of order
).
- The
-Sylow subgroups of
and
are normal in their respective groups, and the fusion system is the same in both cases: In both cases, the action on the subgroup is by the unique subgroup of order
in its automorphism group.
- The fusion systems on the
-Sylow subgroups of
and
are both the inner fusion system: By construction, the
-Sylow subgroup is a retract in both cases (i.e., there is a normal p-complement) so this follows.
The upshot is that and
are fusion system-equivalent.
On the other hand, they are not isomorphic. The reason is that we can recover up to automorphisms inside
from
respectively as follows: first, we recover
as the unique
-Sylow subgroup, then we recover
as
and
as a subgroup of
.
Smallest examples of order 24
We take to get examples of order 24. There are two example pairs:
- direct product of S3 and Z4 with direct product of Dic12 and Z2: These correspond to choosing
as direct product of Z4 and Z2 and considering the two different automorphism classes of subgroups of order 4 in it (Z4 in direct product of Z4 and Z2 and first omega subgroup of direct product of Z4 and Z2).
- dihedral group:D24 and semidirect product of Z3 and D8 with action kernel V4: These correspond to choosing
as dihedral group:D8 and considering the two different automorphism classes of subgroups of order 4 in it (cyclic maximal subgroup of dihedral group:D8 and Klein four-subgroups of dihedral group:D8).