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Then ρ(I)/|I| > 5/2. 6. Let J1 , . . , Jt denote the smallest subgroups among the higher ramication groups with xed intersection with I . Let mi denote the number of terms among the higher ramication groups which intersect I in Ji . Then the RiemannHurwitz formula yields: Proof. ρ(I)/|I| ≥ 1 − 1/|I| + (1/|I|) mi (|Ji | − 1). Now mi is a multiple of |C| and also by HasseArf mi is a multiple of |I : Ij |. Thus, ρ(I)/|I| ≥ 1 − 1/|I| + (1/rm) (1 − 1/|Ji |) ≥ 1 − 1/|I| + t/(2rm), whence the result.

It follows that N is a central product of components Q1 , . . , Qt . By 36 ROBERT GURALNICK minimality, each of the Qi are conjugate in G. Also, we may assume that every minimal normal noncentral subgroup has this form. Since CG (N ) = Z(G), it follows that N is unique. So if t = 1, we see that (S) holds. So assume that t > 1. Since N acts absolutely irreducibly on V , it follows that V = W1 ⊗ . . ⊗ Wt ˆ is a ˆ1 × . . × Q ˆ t ≤ GL(W1 ) × . . × GL(Wt ) where Q ˆi ∼ and N embeds in Q =Q covering group of Qi .

Thus, H ∩ A1 A2 = {(a, φ(a)|a ∈ A1 } for some isomorphism φ : A1 → A2 as required. Thus, we are in case (2). So we may assume that A is the unique minimal normal subgroup of G and A = L1 × . . × Lt with the Li conjugate nonabelian simple groups (and components). If H ∩ A = 1, there is nothing more to say (except to show that t > 1 this requires the classication of nite simple groups in the form of the Schreier conjecture that outer automorphism groups are solvable see [9] for details). Suppose that H1 = H ∩ L1 = 1.