Fusion system induced by a finite group on a finite p-subgroup

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Suppose G is a group (usually finite, though not necessarily so). Suppose P is a subgroup of G that is a finite p-group. The fusion system on P induced via conjugation by G, which we will denote \mathcal{F}_P(G), is a category defined as follows: For any g \in G and R,S \le P such that gRg^{-1} \le S, there is a morphism \varphi:R \to S given by \varphi(r) = grg^{-1}.

Note that this category is a fusion system in the weak sense, but not necessarily a saturated fusion system, which is what many people mean when they talk of fusion system.

Relation between the transporter system and the fusion system

Suppose G is a finite group, p is a prime number, and P is a finite p-subgroup of G. Consider the following two categories:

The object sets of the two categories are identical, but the morphism sets differ. There is a natural "forgetful" functor from \mathcal{T}_P(G) to \mathcal{F}_P(G) defined as follows:

  • Each object of \mathcal{T}_P(G), namely, a subgroup of P, is sent to the same subgroup of P, now viewed as an object of \mathcal{F}_P(G).
  • The morphism set map is as follows: the element g \in N_G(R,S), which is a morphism from R to S in \mathcal{T}_P(G), gets sent to the homomorphism \varphi:R \to S given by x \mapsto gxg^{-1}.