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On finite groups whose 2-Sylow subgroups have cyclic subgroups of index 2*

Published online by Cambridge University Press:  09 April 2009

W. J. Wong
W. J. Wong
Affiliation:
University of Otago.
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If the finite group G has a 2-Sylow subgroup S of order 2a+1, containing a cyclic subgroup of index 2, then in general S may be one of the following six types [8]:

(i) cyclic; (ii) Abelian of type (a, 1), a > 1; (iii) dihedral1; (iv) generalized quaternion; (v) {α, β}, α2a = β2, α2a−1+1, a ≧ 3;

(vi) {α, β}, α2a = β2, α2a−1+1, a ≧ 3.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 4 , Issue 1 , February 1964 , pp. 90 - 112
Copyright
Copyright © Australian Mathematical Society 1964

References

[1]Brauer, R., Zur Darstellungstheorie der Gruppen endlicher Ordnung II, Math. Zeitschr. 72, 25–46 (1959).Google Scholar
[2]Brauer, R., On blocks of representations of finite groups, Proc. Nat. Acad. Sci. U.S.A. 47, 1888–1890 (1961).CrossRefGoogle ScholarPubMed
[3]Brauer, R., On the structure of groups of finite order, Proc. International Congress of Mathematicians, Amsterdam, 1954, vol. I, 209–217.Google Scholar
[4]Bruer, R. and Nesbitt, C., On the modular characters of groups, Annals of Math. 42, 556–590 (1941).Google Scholar
[5]Brauer, R. and Suzuki, M., On finite groups of even order whose 2-Sylow group is a quaternion group, Proc. Nat. Acad. Sci. U.S.A. 45, 1757–1759 (1959).Google Scholar
[6]Dieudonné, J., La géométrie des groups classiques, Berlin, Springer, 1955.Google Scholar
[7]Gorenstein, D. and Waler, J. H., On finite groups with dihedral Sylow 2-subgroups, Illinis Jour. Math. 553–593 (1962).Google Scholar
[8]Hall, M., The theory of groups, New York, Macmillan, 1959.Google Scholar
[9]Hall, P., A note on soluble groups, Jour. London Math. Soc. 3, 98–105 (1928).CrossRefGoogle Scholar
[10]Higman, D. G., Focal series in finite groups, Canadian Jour. Math. 6, 477–497 (1953).CrossRefGoogle Scholar
[11]Schur, I., Über eine Klasse von endlichen Gruppen linearer Substitutionen, Sitzungsber. Preuss. Akad. (1905), 77–91.Google Scholar
[12]Schur, I., Untersuchungen über die Darstellung der endlichen Gruppen durch gebrochene lineare Substitutionen, Jour. für die reine u. angew. Math. 132, 85–137 (1907).Google Scholar
[13]Suzuki, M., Applications of group charcters, Proc. Symposia in Pure Math. I, Providence, R.I., (1959).Google Scholar
[14]Wielandt, H., Über Produkte von nilpotenten Gruppen, Illinois Jour. Math. 2, 611–618 (1958).Google Scholar
[15]Witt, E., Die 5-fach transitiven Gruppen von Mathieu, Abh. Math. Sem. Univ. Hamburg, 12, 256–264 (1938).Google Scholar