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Maximal Abelian Subgroups of the Symmetric Groups

Published online by Cambridge University Press:  20 November 2018

John D. Dixon
John D. Dixon*
Affiliation:
Carleton University, Ottawa, Ontario
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Our aim is to present some global results about the set of maximal abelian subgroups of the symmetric group Sn. We shall show that certain properties are true for “almost all” subgroups of this set in the sense that the proportion of subgroups which have these properties tends to 1 as n → ∞. In this context we consider the order and the number of orbits of a maximal abelian subgroup and the number of generators which the group requires.

Earlier results of this kind may be found in the papers [1; 2; 3; 4; 5]; the papers of Erdös and Turán deal with properties of the set of elements of Sn. The present work arose out of a conversation with Professor Turán who posed the general problem: given a specific class of subgroups (e.g., the abelian subgroups or the solvable subgroups) of Sn, what kind of properties hold for almost all subgroups of the class?

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 23 , Issue 3 , June 1971 , pp. 426 - 438
Copyright
Copyright © Canadian Mathematical Society 1971

References

1. Dixon, J. D., The probability of generating the symmetric group, Math. Z. 110 (1969), 199–205. 438 JOHN D. DIXONGoogle Scholar
2. Erdös, P. and Turán, P., On some problems of a statistical group-theory. II, Acta Math. Acad. Sci. Hun. 18 (1967), 151–163.Google Scholar
3. Erdös, P. and Turán, P., On some problems of a statistical group-theory. III, Acta Math. Acad. Sci. Hung. 18 (1967), 309–320.Google Scholar
4. Erdös, P. and Turán, P., On some problems of a statistical group-theory. IV, Acta Math. Acad. Sci. Hung. 19 (1968) 413–435.Google Scholar
5. Erdös, P. and Turán, P., On some problems of a statistical group-theory. V (to appear).Google Scholar
6. Hardy, G. H. and Wright, E., Introduction to the theory of numbers (Clarendon Press, Oxford, 1954).Google Scholar
7. Kurosh, A. G., Theory of Groups, Vol. 1 (Chelsea, N.Y., 1955).Google Scholar
8. Scott, W., Group Theory (Prentice-Hall, N.J., 1964).Google Scholar