974 i3. MID-SIZE GROUPS OVER F2
fused under Hi and H 2 , H has one class of involutions. Further Cat (z) = J(T) by
13.9.7, so CH(tz) = CH(z) = J(T) nH =TH~ Dg. Therefore by I.4.1, H ~ L3(2)
or A5.
Now Mi:::; Na(E) :::; Na(H), and Mi centralizes 02 (M 2 ) =Hi~ A 4 , so from
the structure of Aut(H), 02 (Mi) :::; Ca(H), and indeed Mi centralizes H if H ~
£3(2). Therefore as m3(MiH) :::; 2 since Na(H) is an SQTK-group, H ~ £ 3 (2) and
MiH =Mix H. But by 13.9.9.3, there is zg E Mi -0^2 (Mi), so L 3 (2) ~ H:::; G~,
contrary to 13.9.8.1. This contradiction completes the proof of Theorem 13.9.1.