In Part II of 3-Transposition groups we prove the existence of the three Fischer groups and show each Fischer group is the unique finite group with a certain involution centralizes Establishing the existence of the Fischer groups supplies a coda to Fischer's Theorem by demonstrating that the sporadic 3-transposition groups of type M(22), M(23), and M(24) appearing in the statement of Fischer's Theorem do indeed exist. It also establishes the existence of these sporadic groups for purposes of the classification of the finite simple groups.

Fischer's Theorem says the Fischer groups are unique as 3-transposition groups with suitable properties, but this is not the right uniqueness result for purposes of the classification. The centralizer of involution characterizations of Part II supplies the appropriate uniqueness results.

The finite simple groups are classified in terms of properties of their local subgroups and particularly centralizers of suitable involutions. See Section 48 in [FGT] and [SG] for a more detailed discussion of the classification and the place of characterizations by involution centralizers in the classification. In particular in the Introduction to [SG] the author's preferred hypothesis, Hypothesis ℌ(w, L), for characterizing sporadic groups is discussed.