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The 26 sporadic groups are investigated with varying emphasis. The 3- transposition groups (Fischer Groups) are only handled in a marginal way, the author plans a separate volume on this subject. The groups $$ON$$, $$Ly$$, $$J_ 1$$, $$J_ 3$$, $$J_ 4$$, $$Ru$$ are not treated seriously at all.
In part I the author studies graphs associated with rank 3-permutation groups, and introduces the notions of geometries in the sense of Tits. He develops as an important group theoretic tool the theory of large extraspecial 2-groups ($$Q$$ is a large extraspecial 2-group in $$G$$ if $$Q$$ is an extraspecial 2-group with $$C_ G(Q) = Z(Q)$$, $$Q\trianglelefteq C_ G(Z(Q))$$, and $$\text{ И}_ G(Q,2') = 1$$). A section on algebras and forms gives the foundation for the later construction of the Griess Algebra. A somewhat more special section treats symplectic 2-loops. Here the Parker Loop (used in the construction of the Monster) is put into a general context. The last section (chap. 5) of part I gives a very readable exposition of the discovery of the sporadic groups and a discussion of existence and uniqueness proofs.
In part III first a method due to Y. Segev and the author is developed. It translates the question whether a group satisfying a certain group theoretic hypothesis into a question on graphs: For this purpose the author introduces a theory of coverings of graphs and complexes. A graph or a complex is called simply connected if it admits no proper coverings. Then the author develops a machinery which associates to the group in question a certain graph. Uniqueness then means that the graph is simply connected. Again this section is of general nature while the following ones are more technical. The first application of the above method is the uniqueness proof of $$M_{24}$$, $$He$$, $$L_ 5(2)$$. A uniqueness proof for $$U_ 4(3)$$ lays ground for uniqueness proofs for $$Co_ 1$$, $$Suz$$, and $$J_ 2$$. The book closes with the derivation of the normalizers of subgroups of prime order in $$M_{24}$$, $$J_ 2$$, $$Co_ 1$$, $$He$$, and $$Suz$$.