11 - The existence and uniqueness of the Fischer groups

Summary

In this chapter we show that there do indeed exist 3-transposition groups of type M(22), M(23), and M(24). In addition we prove the uniqueness of M(22), M(23), and M(24)′ as groups with a suitable involution centralizes Indeed we use the uniqueness results to establish the existence of the Fischer groups.

More precisely in 32.4 in [SG] it is shown that there exists a subgroup X of the Monster such that E(X) is quasisimple with center Z of order 3, X is the split extension of E(X) by an involution inverting Z, and E(X)/Z and X/Z are groups of type F₂₄ and Aut(F₂₄), respectively, in the sense of Sections 34 and 35 or the Introduction to Part II. In Theorem 35.1 we show each group of type Aut(F₂₄) is generated by 3-transposition and is of type M(24). Hence Fischer's three 3-transposition groups exist, and, by Fischer's Theorem, groups of type Aut(F₂₄) are unique up to isomorphism.

We characterize the Fischer groups (and M(24)′; = F₂₄) via the centralizer of an involution by showing each group of type is of type M, for M = M(22) and M(23), and by showing all groups of type F₂₄ are isomorphic. See the Introduction to Part II for an outline of the proofs of these results.