A geodesic insight into some fundamental fusion theorems

Let p be an odd prime and P a Sylow p-subgroup of a finite group G. If P is either metacyclic or each of its elements of order p lies in the center, then \(N_G(P)\) controls strong G-fusion in P, as established in Martino and Priddy (Math. Z. 225(2):277–288, 1997, Theorems 2.7 and 4.1). First, we provide alternative proofs for these results without relying on the Alperin fusion theorem, thereby simplifying the theoretical framework. Second, we establish an equivalence for the control of fusion in terms of a permutation character. Specifically, we define the permutation character induced by the action of G on \(Syl_p(G)\) as the Sylow p-character of G. Now let \(P\in Syl_p(G)\), and \(N_G(P)\le N \le G \). Set \(\chi ,\psi \) to be the Sylow p-characters of G and N, respectively. Then we prove that N controls G-fusion in P if and only if \(\frac{\chi (g)}{\psi (g)}=\frac{|C_G(g)|}{|C_N(g)|} \text { for all } g\in P.\) In the case that N is a p-local subgroup, further results are obtained.