654 5. THE GENERIC CASE: L2(2n) IN .Ct AND n(H) >^1
(5) [T: S[ = 2.
PROOF. Let M2 := M 2 /U. There is a unique T*-invariant subgroup KY,~ A5
of M2, and KY,T* is the global stabilizer in M2 of {6, 7}, so (1) holds. Then VU /U
is the 4-group with fixed-point set {5, 6, 7} and N M 2 (VU) = N M 2 (V) = Mi,2 as
M E M(T), so (2) holds.
Let M 2 ::::; Mo E M(T). By 5.2.8, M2 E £*(G,T), so M2 '.'::l Mo by 1.2.7.3.
Then U = 02 (M2)::::; 02(Mo), so as T::::; M2, U = 02(Mo) by A.1.6. As Mo E 'He,
M 0 /U::::; GL(U), so as M 2 /U is self-normalizing in GL(U), Mo= M2, proving (4).
As V = 02 (LT), 02 (M) = V = Ca(V) by 3.2.11, so M/V ::::; GL(V). Next
UV E Syl 2 (L), so by a Frattini Argument, M = LNM(UV) 2: LNM(U) = LM1,2
using (4). From the structure of M 2 , M 1 , 2 /V is isomorphic to a Borel group of
I'L2(4), so LM1,2/V = NaL(v)(L/V) as NaL(v)(L/V) ~ I'L2(4). Then as L '.'::l M,
(3) holds, and (3) implies (5). D
LEMMA 5.2.12. (1) Z(T) = (z) is of order 2.
(2) Ca(z) = CM 2 (z) is an L3(2)-block.(3) M2 is transitive on U#.
(4) U is a TI-set in G.PROOF. Parts (1) and (3) are easy consequences of the fact that M2 is an
exceptional A 7 -block containing T. As another consequence, Y := CM 2 (z) is an
£3(2)-block. Let Gz := Ca(z) and a;:= Gz/(z). As T::::; Gz, F(Gz) = 02(Gz)
by 1.1.4.6, so F(G;) = 02(Gz)* by A.1.8. Thus as U = 02(Y) 2: 02(Gz) by
A.1.6, and Y is irreducible on U*, U = 02 (Gz)· Thus Gz ::::; Na(U) = M2 using
5.2.11.4. Therefore (2) holds. Then (2), (3), and I.6.1.1 imply (4). D
LEMMA 5.2.13. G has one conjugacy class of involutions.
PROOF. All involutions of V are conjugate under M and hence fused into
Un V. Similarly all involutions in U are conjugate under M 2 , so as U and V are
the maximal elementary abelian subgroups of UV, all involutions in UV are fused
in G. From the structure of M2, each involution in M 2 is fused into UV in M2. So
the lemma holds, as M 2 contains a Sylow 2-group T of G. D
LEMMA 5.2.14. (1) G is transitive on its elements of order 3 which centralize
involutions.(2) All elements of order 3 in Mi U M2 are conjugate in G.
PROOF. By 5.2.12.2, Ca(z) has one class of elements of order 3, so 5.2.13
implies (1). Next M 2 has two classes of elements of order 3, those with either 1 or
2 cycles of length 3 on D. The first class centralizes an involution in M 2 /U and
hence has centralizer of even order. The second class centralizes an involution in U.
Thus (1) implies all elements of order 3 in M 2 are conjugate in G. Then as M 1 , 2
contains a Sylow 3-group of Mi and M 2 , (2) holds. D
LEMMA 5.2.15. Let X E Sy[s(CM(L/0 2 (£)). Then Na(X) = NM(X) ~
I'L2(4).
PROOF. First NM(X) ~ I'L 2 (4). On the other hand by 5.2.14, Xis conjugate
to Y::::; Ca(z) and Ca(Y(z)) = Y x'Cu(Y) ~ Z 3 x E4. Let Gy := Ca(Y) and
Gy := Gy /Y. Then Cay. (z*) ~ E4, and as CM(X) is not 2-closed, neither is