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We prove that in a finite 2-group with no normal Abelian subgroup of rank ä 3, every subgroup can be generated by four elements. This result is then used to determine which 2-groups T with no normal Abelian subgroup of rank ä 3 can occur as 52's of finite simple groups G, under certain assumptions on the embedding of T in G. Introduction. In view of the Feit-Thompson theorem (namely, that all finite groups of odd order are solvable), it is natural to classify and search for finite simple groups by considering the structure of a Sylow 2-subgroup and its embedding in the group. There are comparatively few 2-groups with no normal Abelian subgroup of rank 2—such a 2-group must be cyclic, dihedral, semidihedral, or generalized quaternion. Of these, only dihedral and semidihedral groups can (and do) occur as Sylow 2subgroups of simple groups (Brauer [4, Theorem 2, p. 321]). However, there is a greater range of 2-groups with no normal Abelian subgroup of rank 3 ; this paper is part of a project to determine which such 2-groups can occur as Sylow 2-subgroups of simple groups. The first result is a theorem about 2-groups with no normal Abelian subgroup of rank 3 (apart from questions about simple groups). We show that every subgroup of such a 2-group can be generated by four elements. This is best-possible ; in fact, there are 2-groups satisfying the hypothesis and having elementary subgroups of rank 4. Analogs of the four-generator theorem for odd primes are more restrictive ; for instance, if p is odd and F is a /»-group with no normal Abelian subgroup of rank 3, then every subgroup of F can be generated by three elements (Thompson; Huppert [9, Satz III. 12.3, p. 343]) and every Abelian subgroup of F can be generated by two elements (Feit and Thompson [6, Lemma 8.4(f), p. 797]). Received by the editors September 12, 1969. AMS Subject Classifications. Primary 2029, 2040; Secondary 2025, 2027, 2043.

On 2-groups with no normal abelian subgroups of rank 3, and their occurrence as Sylow 2-subgroups of finite simple groups