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2-groups with few conjugacy classes

Published online by Cambridge University Press:  20 January 2009

Nigel Boston  and
Judy L. Walker
Nigel Boston
Affiliation:
Department of Mathematics, University of Illinois, Urbana, IL 61801, USA ([email protected])
Judy L. Walker
Affiliation:
Department of Mathematics and Statistics, University of Nebraska, Lincoln, NB 68588, USA ([email protected])
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Abstract

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An old question of Brauer that asks how fast numbers of conjugacy classes grow is investigated by considering the least number cn of conjugacy classes in a group of order 2n. The numbers cn are computed for n ≤ 14 and a lower bound is given for c15. It is observed that cn grows very slowly except for occasional large jumps corresponding to an increase in coclass of the minimal groups Gn. Restricting to groups that are 2-generated or have coclass at most 3 allows us to extend these computations.

Keywords

p-groupsconjugacy classescoclass
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 43 , Issue 1 , February 2000 , pp. 211 - 217
Copyright
Copyright © Edinburgh Mathematical Society 2000

References

1.Bertram, E. A., On large cyclic subgroups of finite groups, Proc. Am. Math. Soc. 56 (1976), 63–66.Google Scholar
2.Bosma, W. and Cannon, J. J., Handbook of Magma functions (School of Mathematics and Statistics, University of Sydney, 1996).Google Scholar
3.Brauer, R., Representations of finite groups. Lectures on Modern Mathematics (ed. Saaty, T. L.), vol. I (Wiley, New York, 1963).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.CrossRefGoogle Scholar
5.Fernández-Alcober, G. A. and Shepherd, R. T., On the order of p-groups of abundance zero, J. Algebra 201 (1998), 392–400.CrossRefGoogle Scholar
6.Higman, G., Enumerating p-groups, I, Inequalities, Proc. Lond. Math. Soc. (3) 10 (1960), 24–30.CrossRefGoogle Scholar
7.Kovaćs, L. G. and Leedham-Green, C. R., Some normally monomial p-groups of maximal class and large derived length, Q. J. Math. Oxford 37 (1986), 49–54.CrossRefGoogle Scholar
8.Landau, E., Klassenzahl binärer quadratischer Formen von negativer Discriminante, Math. Ann. 56 (1903), 674–678.CrossRefGoogle Scholar
9.López, A. V. and López, J. V., Classification of finite groups according to the number of conjugacy classes, Isr. J. Math. 51 (1985), 305–338.CrossRefGoogle Scholar
10.López, A. V. and López, J. V., Classification of finite groups according to the number of conjugacy classes, II, Isr. J. Math. 56 (1986), 188–221.Google Scholar
11.Newman, M. F. and O'brien, E. A., Classifying 2-groups by coclass, Trans. Am. Math. Soc. 351 (1999), 131–169.CrossRefGoogle Scholar
12.O–brien, E. A., The p-group generation algorithm, J. Symbolic Comput. 9 (1990), 677–698.CrossRefGoogle Scholar
13.Poland, J., Two problems on finite groups with k conjugate classes, J. Aust. Math. Soc. 8 (1968), 49–55.CrossRefGoogle Scholar
14.Pyber, L., Finite groups have many conjugacy classes, J. Lond. Math. Soc. (2) 46 (1992), 239–249.CrossRefGoogle Scholar
15.Shalev, A., The structure of finite p-groups: effective proof of the coclass conjectures, Invent. Math. 115 (1994), 315–345.CrossRefGoogle Scholar
16.Sherman, G. J., A lower bound for the number of conjugacy classes in a finite nilpotent group, Pac. J. Math. 80 (1979), 253–254.CrossRefGoogle Scholar