Pushing up in Finite Groups

The main result is a pushing up theorem for L3(2n). Let G be a finite group with F*(G) = O2(G). Suppose there exists a normal subgroup H of G with O2(G) ≤ H such that H/O2(G) has no partial complement in G/O2(G) and G/H is isomorphic to L3(2n). Let T be a Sylow 2-subgroup of G and let X be a maximal 2-local subgroup of G containing H with XH/H a maximal parabolic subgroup of G/H. Let P = O2(X) and E = Ω1 (Z(Q)). Theorem. If G ≠ {CG(Ω1(Z(T)), NG(J(P))}, then wither (i) [G,Eα] ≤ E for each automorphism α of P or (ii) G has exactly one noncentral chief factor within O2(G). We also derive some new pushing up theorems for L2(2n) which are variations on the results of Baumann, Glauberman and Niles.