Ž . JOURNAL OF ALGEBRA 179, 894904 1996 ARTICLE NO. 0042 Finite Groups Containing Many Subgroups Are Solvable Marcel Herzog* School of Mathematical Sciences, Raymond and Be  erly Sackler Faculty of Exact Sciences, Tel-A i  Uni  ersity, Tel-A i  , Israel Patrizia Longobardi and Mercede Maj Dipartimento di Matematica e Appl.ni, Uni  ersita degli Studi di Napoli, ` Monte S. Angelo,  ia Cinthia, 80126 Naples, Italy and Avinoam Mann* Einstein Institute of Mathematics, The Hebrew Uni  ersity, Gi  at Ram, Jerusalem 91904, Israel Communicated by Walter Feit Received November 16, 1994 I. INTRODUCTION In this paper G denotes a finite group. As is well known, the converse of Lagrange’s theorem in group theory does not hold. That is, given a finite group G of order n, and given a divisor d of n, G need not have a subgroup of order d. Indeed, a celebrated theorem of P. Hall states that it suffices to assume the existence of subgroups of special orders, namely of p-complements, in order to force the group to be solvable. Using the classification of the finite simple groups, this can be strengthened. Thus Z.   Arad and M. B. Ward showed AW that if G contains a 2-complement and a 3-complement, then G is solvable. They also showed that G is  4 solvable if it contains a p, q -Hall subgroup for each pair of primes *These authors are grateful to the Department of Mathematics of the University of Napoli for its hospitality while this investigation was carried out. Their visits were supported by CNR grants. 894 0021-869396 $12.00 Copyright  1996 by Academic Press, Inc. All rights of reproduction in any form reserved.