1.2. THE SET .C.* (G, T) OF NONSOLVABLE UNIQUENESS SUBGROUPS 503Next assume L/Z(L) is of Lie type and odd characteristic; then L* appears
in conclusion (2) of Theorem C. If L* ~ L 2 (p^2 ) then as L 2 (9) ~ Sp 4 (2)', we may
assume p > 3. Therefore as z* · E Z(R*), either z* E L* and O(CL* (z*)) =f=. 1or z induces a field automorphism on L so that CL (z) has a component; in
either case, this is contrary to F(CL·(z)) = 02(CL(z)). The same argument
eliminates L2(P) unless pis a Fermat or Mersenne prime, which is allowed in (e);
as before, the fact that z E Z(R) rules out the double covers SL 2 (p), the onlypossibilities with Z(L) =f=. 1 by I.1.3. Similarly if L ~ LHp) then as z E Z(R*),
z E L; then unless p = 3, CL(z) has an SL 2 (p)-component, for our usual
contradiction. Finally U3(3) ~ G2(2)' was covered earlier, while if L* ~ L 3 (3) thenZ(L) = 1 by I.1.3, so conclusion (e) holds.
This leaves the case L* sporadic, so L* appears in conclusion (5) of Theorem
C. First Ji is ruled out by the existence of a component in CL* (z*). Then as usualz E L since z E Z(R), so that (f) holds.
This completes the proof of 1.1.5. DLEMMA 1.1.6. Let B be a nontrivial 2-subgroup of G, H:::; G with BCc(B) :::;
H:::; Nc(B), SE Syl2(H), Ta Sylow 2-subgroup of G containing S, z an involution
in Z(T), and ME M(Cc(z)). Then the hypotheses of 1.1.5 are satisfied.PROOF. As z E Z(T), M E M(T), so M E He since G is of even characteristic.
Thus as S:::; T, ME 1-le(S). Next B:::; 02 (H):::; S:::; M so that B:::; 02 (H n M),
and henceCo2(M)(02(H n M)):::; Cc(B):::; H.
Also z E CT(B):::; T n H =Sas SE Syh(H), so z E Z(S); hence the hypotheses
of 1.1.5 are satisfied. The proof is complete. D1.2. The set C*(G, T) of nonsolvable uniqueness subgroups
In this section we use our results on the structure of SQTK-groups in section A.3
to establish tools for working in 2-local subgroups of G; such appeals are possible
since our 2-locals are strongly quasithin. In particular we obtain a description
of H^00 for H E 7-l, and also properties of the poset of perfect members of 1-l,
partially ordered by inclusion. Such results then lead to the existence of uniqueness
subgroups of G.
We begin by recalling Definition A.3.1 which defines C-components: For H :::; G,
let C(H) be the set of subgroups L:::; H minimal subject to
1 =f=. L = L^00 ~ ~ H.
The members of C(H) are the C-components of H. As we will see, usually we can
expect there will be HE'}-{ with C(H) nonempty.
We recall also that the elementary results in A.3.3 hold for arbitrary finite
groups. By contrast, the later results in section A.3 requiring Hypothesis A.3.4
apply only to an SQTK-group X with 02(X) = l. We apply those results to
H/0 2 (H) for HE'}-{, and then pull them back to obtain results about H.