5.2. USING WEAK BN-PAIRS AND THE GREEN BOOK 655so CM(X) = Ca(X). Then as INM(X) : CM(X)I = 2 = IAut(X)I, the lemma
follows. D
Recall the definition of the subgroups G 1 and G 2 in our amalgam a from theprevious subsection, and let Go := (G1, G 2 ).
LEMMA 5.2.16. (1) Go~ L3(4).
(2) GoT is Go extended by a field automorphism.
PROOF. Notice in the L 3 (4)-amalgam that we have B = D = BD. Thus toprove (1), it suffices by F.4.26 to show that there exist involutions Si E NLi (B), such
that ls1s2I :::; 3. Then (2) follows from (1), since Tacts on Gi, with IT: SI = 2 by
5.2.11.5, and 02(LiT) = 02(Li) by 5.2.7.3. Thus it remains to exhibit involutions
Si E NLi(B), with ls1s2I:::; 3.Notice that each involution Si E NLi(B) inverts B. Now B :::; M 1 , so by
5.2.14.2, B is conjugate to the subgroup X defined in 5.2.15. Therefore as s 1
inverts B, Na(B) =(Bx LB)(s1), where LB~ A5, s1 inverts B, and s1 induces atransposition on LB· But s2 also inverts B, so replacing s 1 by a suitable member
of Bs1, we may assume s1s2 E LB· Thus s1 and s 2 are distinct transpositions in
LB(s1) ~ 85, so ls1s2I = 2 or 3, completing the proof. D
We now define some notation to use in our identification of G with M 23. Let
G := M23 act on 8 := {1, ... , 23}. Then (cf. chapter 6 in [Asc94]) we may take
our 7-set D to be a block in. the S.teiner system ( e, C) on e preserved by G, sothat N 0 (D) = M2 is the split extension of tJ = Gn ~ E16 by A1, and M2 is an
exceptional A1-block.LEMMA 5.2.17. There is a permutation equivalence ( : M2 --+ M2 of M2 and
M 2 on n.
PROOF. As B is of order 3 in Kn M1,2, it follows from parts (1) and (2)of 5.2.11 that we may choose B to act on D as ((1, 2, 3)). Then as Cu(B) = 1,
NM 2 (B) ~ Z 2 /(Z 3 x A4) has Sylow 2-groups of order 8. Thus T splits over U, so
M 2 splits over U by Gaschiitz's Theorem A.1.39. Thus there is an isomorphism
( : M 2 --+ M 2 , and adjusting by a suitable inner automorphism, this map is a
permutation equivalence. D
For the remainder of this section, define ( as in 5.2.17.
Let r := 92 be the set of unordered pairs of elements from 8 and fix x := {6, 7}
and y := {5, 6} in r. From chapter 6 of [Asc94]:LEMMA 5.2.18. {1) Gx is the extension of L3(4) by a field automorphism.
(2) 8 - { 6, 7} is a projective plane over F 4 with lines { C - { 6, 7} : { 6, 7} s;;:
CE C}, and Gx preserves this structure.
(3) The global stabilizer I of { 4, 5, 6, 7} in G is the global stabilizer in M 2 of
{4,5,6, 7}.
PROOF. In [Asc94], the Steiner system (8, C) is constructed so that (1) and
(2) hold. As each 4-point subset of 8 is contained in a unique block of the Steiner
system, (3) holds. D
Regard I' as a a graph by decreeing that a, b E I' are adjacent if la n bl = 1.
We wish to show G ~ G. To do so, we write Gx for G 0 T and essentially show
there is a graph structure on I'a := G/Gx isomorphic to the graph r, such that the