For the proof of the main theorem of this chapter, the Hall-Wieland t Theorem (see [Ha], Theorem 14.4.2) is needed. Let G be a finite group, p a prime number and P a Sylow p-subgroup of G. A subgroup A of P is said to be weakly closed in P relative to G if, for all g ϵ G, Ag ⊂ P implies that Ag = A. Let Zp−1(P) be the (p − 1)st term of the upper central series of P. Let A be a weakly closed subgroup of P relative to G. The Hall-Wielandt Theorem states that, if A ⊂ Zp−1(P), or if p > 2 and A is abelian, then G/Op(G) is isomorphic to NG(A)/Op(NG(A)).

In this chapter it will be assumed that

(Bl) The subgroup V contains a subgroup P of prime order p such that C G (P) has 2-rank 1.

We will demonstrate the following theorem.

Theorem B.The conclusion of Theorem A holds for G under the hypothesis (Bl).

If G has a normal subgroup of index p, the conclusion of the theorem holds by Chapter I, § 3, Proposition 2. It will be assumed therefore that

(B2)G has no normal subgroup of index p.