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On the laws of certain finite groups

Published online by Cambridge University Press:  09 April 2009

John Cossey ,
Sheila Oates MacDonald  and
Anne Penfold Street
John Cossey
Affiliation:
Department of Pure MathematicsSchool of General StudiesAustralian National UniversityCanberra
Sheila Oates MacDonald
Affiliation:
Department of MathematicsUniversity of QueenslandBrisbane
Anne Penfold Street
Affiliation:
Department of MathematicsUniversity of QueenslandBrisbane
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In recent years a great deal of attention has been devoted to the study of finite simple groups, but one aspect which seems to have been little considered is that of the laws they satisfy. In a recent paper [3], the first two of the present authors gave a basis for laws of PSL(2, 5). The techniques of [3] can be used to show that (modulo certain classification problems) a basis for the laws of PSL(2, pn) can be made up from laws of the following types:

(1) an exponent law,

(2) laws which determine the Sylow subgroups,

(3) laws which determine the normalisers of the Sylow subgroups,

(4) in certain special cases, laws which determine subvarieties of smaller exponent, e.g. the subvariety of exponent 12 for those PSL(2, pn) which contain S4,

(5) a law implying local finiteness.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 11 , Issue 4 , November 1970 , pp. 441 - 489
Copyright
Copyright © Australian Mathematical Society 1970

References

[1]Burnside, W., Theory of Groups (2nd edition, 1911, Dover, New York, 1955).Google Scholar
[2]Cossey, John, On varieties of A-groups (Ph. D. dissertation, Australian National University, 1966).Google Scholar
[3]Cossey, John and MacDonald, Sheila Oates, ‘A basis for the laws of PSL(2, 5)’, Bull. Amer. Math. Soc. 74 (1968), 602–606.CrossRefGoogle Scholar
[4]Cossey, John, MacDonald, Sheila Oates and Street, Anne Penfold, ‘On the laws of certain linear groups’, Bull. Amer. Math. Soc. 75 (1969), 361–363.CrossRefGoogle Scholar
[5]Curtis, C. W. and Reiner, I., Representation theory of finite groups and associative algebras (Interscience, New York, 1962).Google Scholar
[6]Dickson, L. E., Linear groups, (1900, Dover, New York 1958).Google Scholar
[7]Gaschütz, W., ‘Zur Erweiterungstheorie der Endlichen Gruppen’, J. reine angew. Math. 190 (1952), 93–107.CrossRefGoogle Scholar
[8]Gorenstein, D., Finite groups (Harper and Row, New York, 1968).Google Scholar
[9]Hall, M., ‘Solution of the Burnside problem for exponent 6’, Illinois J. of Math. 2 (1958) 764–786.CrossRefGoogle Scholar
[10]Hall, M., The theory of groups (Macmillan, New York, 1959).Google Scholar
[11]Hall, P. and Higman, G., ‘On the p-length of p-soluble groups and reduction theorems for Burnside's problem’, Proc. London Math. Soc. (3) 6 (1956), 1–43.CrossRefGoogle Scholar
[12]Higman, G., ‘Complementation of abelian normal subgroups’, Publ. Math. Debrecen, 4 (1956), 455–458.CrossRefGoogle Scholar
[13]Huppert, B., Endliche Gruppen I (Die Grundlehrender mathematischen Wissenschaften, Bd. 134. Springer-Verlag, Berlin, 1967).CrossRefGoogle Scholar
[14]Köchendorffer, R., ‘Über treue irreduzible Darstellungen endlicher Gruppen’, Math. Nachr. (1948), 25–39.CrossRefGoogle Scholar
[15]Kovács, L. G. and Newman, M. F., ‘Cross varieties of groups’, Proc. Roy. Soc. (London) A, 292 (1966), 530–536.Google Scholar
[16]Kovács, L. G. and Newman, M. F., ‘On critical groups’, J. Austral. Math. Soc. 6 (1966), 237–250.CrossRefGoogle Scholar
[17]Neumann, Hanna, Varieties of groups (Erbegnisse der Mathematik und ihrer Grenzgebiete, Bd. 37, Springer-Verlag, Berlin, 1967).CrossRefGoogle Scholar
[18]Oates, Sheila, ‘Identical relations in groups’, J. London Math. Soc. 38 (1963) 71–78.CrossRefGoogle Scholar
[19]Oates, Sheila and Powell, M. B., ‘Identical relations in finite groups’, J. Algebra 1 (1964), 11–39.CrossRefGoogle Scholar
[20]Schur, I., ‘Untersuchungen über die Darstellung der endlichen Gruppen durch gebrochene lineare Substitutionen’, J. reine angew. Math. 132 (1907), 85–137.Google Scholar
[21]Taunt, D. R., ‘On A-groups’, Proc. Cambridge Phil. Soc. 45 (1949), 24–42.CrossRefGoogle Scholar
[22]Walter, J. H., ‘The characterisation of finite groups with abelian Sylow 2-subgroups’, (to appear).Google Scholar
[23]Weichsel, P. M., ‘A decomposition theory for finite groups with applications to p-groups’, Trans. Amer. Math. Soc. 102 (1962), 218–226.Google Scholar
[24]Weichsel, P. M., ‘On critical p-groups’, Proc. London. Math. Soc. (3) 14 (1964), 83–100.CrossRefGoogle Scholar