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Finite Groups Containing Many Subgroups Are Solvable

✍ Scribed by Marcel Herzog; Patrizia Longobardi; Mercede Maj; Avinoam Mann

Publisher: Elsevier Science
Year: 1996
Tongue: English
Weight: 160 KB
Volume: 179
Category: Article
ISSN: 0021-8693
DOI: 10.1006/jabr.1996.0042

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✦ Synopsis

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. w x 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 Ä 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

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