Subgroups of Groups of Monster Type
Michael Aschbacher · Cambridge University Press eBooks · 1994
In Chapter 10 we constructed a finite simple group G possessing an involution z such that F *( C G (Z )) = Q is extraspecial of order 2 1+24 , C G (Z)/Q ≅ Co 1 , Q/〈z〈 is isomorphic to the Leech lattice modulo 2 as a C G ( z )/Q-module, and z is not weakly closed in Q with respect to G . We say that a group G satisfying these hypotheses is of Monster type . In this short chapter we investigate groups of Monster type. In particular we see that such a group contains simple subgroups of type F p , for p = 2, 3, 5, 7, and 24. See Section 32 for the definition of groups of type F p . Now from Chapter 5, there are twenty-six sporadic groups. In Chapter 6 we constructed the five Mathieu groups. In Chapters 8 and 9 we constructed the three Conway groups plus Suz , J 2 , HS , and Mc . Hence each of these twelve sporadic groups is a section of the Monster. Similarly the sporadic groups F 1 , F 2 , F 3 , and F 5 are of type F p , and hence sections of the Monster. Held's group He is of type F 7 and the largest Fischer group F 24 = μ(24)′ is of type F 24 , so these groups are sections of the Monster. Finally the Fischer groups μ(22) = F 22 and μ(23) = F 23 are sections of F 24 , and hence also of the Monster. This is best seen by viewing Aut ( F 24 ) = μ(24) as a 3-transposition group. Thus we have existence proofs for twenty of the twenty-six sporadic groups.