Fusion system-equivalent not implies isomorphic

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Statement

It is possible to have two fusion system-equivalent finite groups G_1 and G_2 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 p-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 p,q such that p divides q - 1. Let A be the cyclic group of order q. Consider a group B of order p^3 that has more than one equivalence class up to automorphisms of subgroups of order p^2. Let C_1,C_2 be representatives of two distinct equivalence classes of such subgroups.

Now define:

G_1 = A \rtimes B

where the action of B on A is given by a homomorphism B \to \operatorname{Aut}(A) with kernel C_1. Note that the image is the unique cyclic subgroup of order p inside \operatorname{Aut}(A), which is cyclic of order q - 1.

Also define:

G_2 = A \rtimes B

where the action of B on A is given by a homomorphism B \to \operatorname{Aut}(A) with kernel C_2. Note that the image is the unique cyclic subgroup of order p inside \operatorname{Aut}(A), which is cyclic of order q - 1.

We note that:

  • Both G_1 and G_2 have order p^3q. Moreover, the q-Sylow subgroups of both are isomorphic to A (cyclic of order q) and the p-Sylow subgroups are isomorphic to B (of order p^3).
  • The q-Sylow subgroups of G_1 and G_2 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 p in its automorphism group.
  • The fusion systems on the p-Sylow subgroups of G_1 and G_2 are both the inner fusion system: By construction, the p-Sylow subgroup is a retract in both cases (i.e., there is a normal p-complement) so this follows.

The upshot is that G_1 and G_2 are fusion system-equivalent.

On the other hand, they are not isomorphic. The reason is that we can recover C_i up to automorphisms inside B from G_i respectively as follows: first, we recover A as the unique q-Sylow subgroup, then we recover B as G_i/A and C_i = C_G(A)/A as a subgroup of B.

Smallest examples of order 24

We take p = 2, q = 3 to get examples of order 24. There are two example pairs: