16.1. EVEN TYPE GROUPS, AND COMPONENTS IN CENTRALI:ZERS 1171NOTATION 16.1.3. Recall that the types of twisted groups in Theorem C are
(^2) A2 = U3, (^2) B2 = 8z, (^3) D4, and (^2) F4· We adopt the convention of [GLS98] for
labeling involutory outer automorphisms of these groups. We emphasize that this
convention differs from that of Steinberg which is widely used in the literature
( eg. in the Atlas [C+s5]) in which there are no graph automorphisms of twisted
groups. Instead the convention in Definition 2.5.3 in [GLS98] is that all involutory
automorphisms of groups of type^2 A 2 = U 3 which are not inner-diagonal are calledgraph automorphisms, but involutory outer automorphisms of groups of type^3 D 4
are called field automorphisms. All involutory automorphisms of groups of types(^2) B 2 = Sz and (^2) F4 are inner.
LEMMA 16.1.4. Assume that L ~ X(2n) is of Lie type X and characteristic
2. Let r be an involution in Aut(L), and set Lr := 02 (CL(r)). Then one of the
following holds:
(1) r induces an automorphism on L corresponding to a root involution of L
(or in 8p4(2) or G 2 (2), if L = 8p4(2)' or G2(2)'), and Lr = 02 (Cp(r)) for the
proper parabolic P containing CL( r).
(2) L ~ Sp4(2n), r induces an automorphism of type c2, and Lr= 1.(3) L ~ L4(2) or Ls(2), r induces an automorphism of type J2, and Lr ~ A4
or Z 3 /(E4 x E4), respectively.
(4) r induces a field automorphism on L and Lr~ X(2n/^2 )^1. Further 8z(2n)
and^2 F 4 (2n) have no involutory non-inner automorphisms, and U3(2n) has no in-
volutory field automorphisms.
(5) L ~ L 3 (2n), n even, r induces a graph-field automorphism on L, andLr~ U3(2nl^2 )-unless n = 2, where Lr~ Eg.
( 6) L ~ L3( 2n), r induces a graph automorphism on L, and Lr ~ L2 ( 2n )'.
(7) L ~ 8p 4 (2n), n odd, r induces a graph-field automorphism on L, and
Lr~ 8z(2n)'.
(8) L ~ L 4 (2) or L 5 (2), r induces a graph automorphism on L, and Lr~ A5.(9) L(r) ~ 8s, r is of type 23 , 12 , and Lr~ A4·
PROOF. This follows from list of possibilities for L in (E2), and the 2-localstructure of Aut(L) (cf. Aschbacher-Seitz [AS76a]). D
We turn to the cases in parts (ii) and (iii) of (E2):
LEMMA 16.1.5. Assume that L is not of Lie type and characteristic 2, and r is
an involution in Aut(L). Then one of the following holds:
(1) L ~ L 3 (3) and either r is inner with CL(r) ~ GL2(3), or r is outer with
CL(r) ~ 84.(2) L ~ L 2 ( q) for q > 7 a Fermat or M ersenne prime, and either r is inner with
CL(r) E 8yb(L), or r is an outer automorphism in PGL2(q) with CL(r) ~ Dq+E,
where q = E mod 4.
(3) L ~ M 11 , r is inner, and CL(r) ~ GL2(3).
(4) L ~ M 12 and either r is inner with CL(r) ~ 83/Q~ or Z2 x 85, or r isouter and C1(r) ~ Z2 x As.
(5) L ~ M 22 and either r in inner with CL(r) ~ 84/E15, or r is outer with
C1(r) ~ L3(2)/Es or 8z(2)/E15.
(6) L ~ M2 3 , r is inner, and CL(r) ~ L3(2)/E15.
(7) L ~ M24, r is inner, and CL(r) ~ L3(2)/D~ or 8s/E54.