# 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).