7 - Some group extensions

Summary

In this chapter we investigate certain extensions of groups. We begin in Section 21 with a general discussion of the 2-cohomology group H²(G, V) of the representation of a group G on an abelian group V. The 2-cohomology group keeps track of the number of isomorphism types of extensions Ĝ of V by G in which the induced representation of Ĝ/V on V is equivalent to that of G on V. We recall the standard theory of 2-cohomology in Section 21. The result we need is 21.8, which guarantees that if G₁ and G₂ are two such extensions with V an abelian p-group, G faithful on V, and H_i a subgroup of G_i with |G_i : H_i prime to p, then under suitable conditions an isomorphism φ : H₁ → H₂ extends to an isomorphism of G₁ with G₂.

In Section 22 we recall facts about the Todd modules V for the Mathieu groups G found in [SG]. Then we go on to determine the 1-cohomology group H¹(G, V) of the Todd modules. The 1-cohomology group has at least three important group theoretic interpretations; see for example Section 17 in [FGT]. We will use the results on the 1-cohomology of the Todd modules in proving the uniqueness of the Fischer groups subject to suitable hypotheses. In particular we use such results in Section 23 to determine the perfect central extensions of certain sections of the Fischer groups. We also determine the conjugacy classes of elements of order 2 and 3 in U₆(2) in Section 23.