On the Sylow graph of a group and Sylow normalizers

Let G be a finite group and G (p) be a Sylow p-subgroup of G for a prime p in pi(G), the set of all prime divisors of the order of G. The automiser A (p) (G) is defined to be the group N (G) (G (p) )/G (p) C (G) (G (p) ). We define the Sylow graph I" (A) (G) of the group G, with set of vertices pi(G), as follows: Two vertices p, q a pi(G) form an edge of I" (A) (G) if either q a pi(A (p) (G)) or p a pi(A (q) (G)). The following result is obtained Theorem: Let G be a finite almost simple group. Then the graph I" (A) (G) is connected and has diameter at most 5. We also show how this result can be applied to derive information on the structure of a group from the normalizers of its Sylow subgroups.