Characters of Abelian Groups by Paley R.E.A.C., Wiener N.

By Paley R.E.A.C., Wiener N.

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Since CV (C) = 1 and [W, C] ≤ V the restriction of λ to W is an isomorphism. Furthermore, since V is an abelian normal subgroup in W , λ(W ) ≤ CA (V ) and since W/V is a trivial C-module, λ(W ) ≤ CA (G/V ). Hence λ(W ) ≤ D(C, V ) and composing the restriction of λ to W with ϕ−1 we obtain the required injection ψ. 2). 1), so that Cγ = {cγ(c) | c ∈ C} is a complement to Q in G. Suppose further that γ(c) ∈ Z(Q) for every c ∈ C. Then dγ : cq → cγ(c)q is a deck automorphism of G which centralizes Q and maps C onto Cγ .

Respectively, we also have the following. 5 Ω+ 6 (2) = Alt8 = L4 (2). 10 L2 (7) and L3 (2) Let P8 = {∞, 0, 1, . . , 6} be the projective line over the field GF (7) of seven elements. The stabilizer of the projective line structure in the symmetric group of P8 is P GL2 (7) and the latter contains the group L2 (7) with index 2. 9 provides us with an embedding P GL2 (7) ≤ O6+ (2). In this section we refine this embedding geometrically. 1 Let (V, f, q) = Q(P8 ) and let O6+ (2) be the corresponding orthogonal group.

The exact order is known in the literature (cf. 3 in Aschbacher (1986) in terms of Alt8 and Bell (1978) in terms of L4 (2)). 10 to observe the following. Let L ∼ = L2 (7) act on P8 as on the projective line over GF (7). Then H8 ∼ = I1 ⊕ I2 where I1 and I2 are the natural and the dual natural L3 (2)-modules of L. 3 Let J2 be the preimage of I2 in P8e and put I1 = P8 /J2 . Then (i) I1 is an indecomposable extension of I1 by a 1-dimensional trivial module; (ii) H 1 (L, I1 ) ∼ = 2. Proof Let S be a Sylow 2-subgroup in L.

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