Sylow Centralizers — the Imprimitive Case

In the next two sections, we study a situation where a solvable group G acts faithfully and irreducibly on a finite vector space V and each ν ∈ V is centralized by a Sylow p-subgroup (for a fixed prime divisor p of |G|). The basic thrust is to show that the examples given in 9.1 and 9.4 are essentially the only ones. If V is a quasi-primitive G-module, we show in Section 10 that G ≤ Γ(V) (compare with Example 9.1) or G ≤ GL(2,3) and |V| = 32. In this section we assume that G is imprimitive and Op′ (G) = G. Choose C ⊴ G maximal such that VC is not homogeneous and write VC = V1 ⊕…⊕ Vn for homogeneous components Vi of VC. The main result (Theorem 9.3) employs Gluck's result in Section 5 to show that n = 3, 5 or 8, G/C ≅ D6, D10 or AΓ(23) and p is 2, 2 or 3 (respectively). Furthermore, C transitively permutes the non-zero vectors of Vi for each i. Then Huppert's results of Section 6 apply and C/CC(Vi) ≤ Γ(Vi) unless |Vi| is one of six values. The remainder of this section, somewhat technical, exploits these facts to give detailed information about the normal structure of G.

Example. Let q, p be primes and n an integer such that p | qn − 1. Let V be an n-dimensional vector space over GF(q). Suppose that H ⊴ Γ(V) = Γ(qn) and p | |H|. If ν ∈ V, then CH(v) contains a Sylow p-subgroup of H (and of course of OP′(H)). Also, OP′(H) acts irreducibly on V.