Fusion system induced by a finite group on its p-Sylow subgroup is the inner fusion system iff the group is p-nilpotent

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Suppose G is a finite group and P is a p-Sylow subgroup of G. Then, the fusion system induced by G on P (i.e., \mathcal{F}_P(G)) equals the inner fusion system \mathcal{F}_P(P) if and only if G is a P-nilpotent group (?), i.e., G contains a Normal p-complement (?).

Facts used

  1. Conjugacy-closed and Sylow implies retract


Given: A finite group G, a p-Sylow subgroup P of G, such that \mathcal{F}_P(G) = \mathcal{F}_P(P).

To prove: G has a normal p-complement.


  1. Suppose a,b \in P are conjugate in G. Then, a,b are conjugate in P: Let Q = \langle a \rangle, R = \langle b \rangle. Then, if gag^{-1} = b consider the map \varphi:Q \to R given by \varphi(x) = gxg^{-1}. This is a morphism in \mathcal{F}_P(G), so it is also a morphism in \mathcal{F}_P(P). In particular, there exists h \in P such that hah^{-1} = b.
  2. P has a normal complement in G, so G has a normal p-complement: This follows from the previous step and fact (1).