On the Sylow graph of a group and Sylow normalizers

Abstract

Let G be a finite group and G p be a Sylow p-subgroup of G for a prime p in π(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 Γ A (G) of the group G, with set of vertices π(G), as follows: Two vertices p, q ∈ π(G) form an edge of Γ A (G) if either q ∈ π(A p (G)) or p ∈ π(A q (G)). The following result is obtained: * The second and third authors have been supported by Proyecto MTM2007-68010-C03-03, Ministerio de Educación y Ciencia and FEDER, Spain. The first author thanks the Universitat de València and the Universidad Politécnica de Valencia for their warm hospitality during the preparation of this paper.Theorem: Let G be a finite almost simple group. Then the graph Γ 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.

A survey on Sylow normalizers and classes of groups

“…Similar results for other particular examples appear in [16]. However, we emphasize the following facts in the finite universe E:…”

“…In this direction relevant results appear in relation with subgroup lattices, factorized groups, formations and Fitting classes (see [1] for an account of this development). A further insight appears in [9,16,17] in relation with Sylow normalizers. Again a convenient restriction on the sets of primes locally defining a nilpotent-like Fitting formation leads to the so-called covering formations, which are classes of groups with nilpotent Hall subgroups for adequate sets of primes and play an important role in relation with Sylow normalizers (see Section 4).…”

“…A main goal in the course of this research is the connectedness of the so-called Sylow graph of a group [16,Main Theorem].…”

“…In particular, this is always true if G is a finite simple group (see [15]). More precise information is obtained in [16] in graph terms for almost simple groups.…”

“…Example] Let p ∈ P \ {2, 3, 5, 7, 13}, π = {2, 5, 7, 13, p} and H = LF (h) given by h(q) = E π if q ∈ π\{p}, h(p) = S π , h(t) = ∅ if t / ∈ π. Then H is not a covering formation but n H = H. The class H × E π = (G | G = A × B, A ∈ H, B ∈ E π ) gives an example with full characteristic; see [16,Lemma 5.7]. Proposition 3.2 and Theorem 3.1 in [17] give conditions to guarantee that a subgroup-closed saturated formation H satisfying n H = H is a covering formation, in terms of the canonical local definition of H.…”