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Finite Groups with a given Number of Conjugate Classes

Published online by Cambridge University Press:  20 November 2018

John Poland
John Poland*
Affiliation:
Department of Mathematics, Institute of Advanced Studies, Australian National University, Canberra, and Carleton University, Ottawa
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This paper presents a list of all finite groups having exactly six and seven conjugate classes and an outline of the background necessary for the proof, and gives, in particular, two results which may be of independent interest. In 1903 E. Landau (8) proved, by induction, that for each the equation

*

has only finitely many solutions over the positive integers.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 20 , 1968 , pp. 456 - 464
Copyright
Copyright © Canadian Mathematical Society 1968

Footnotes

This material formed part of the author's dissertation, McGill University, May, 1966.

*

I thank my director H. Schwerdtfeger for calling my attention to Landau's theorem, from this reference.

References

1. Brauer, R., Representations of finite groups, Lectures on Modern Mathematics, éd. T. L. Saaty. Vol. I (New York, 1963).Google Scholar
2. Brauer, R., Some applications of the theory of blocks of characters offinite groups. I, J. Alg., 1 (1964), 152–167.Google Scholar
3. Brauer, R. and Fowler, K. A., On groups of even order, Ann. of Math., 62 (1955), 565–583.Google Scholar
4. Burnside, W., Theory of groups of finite order (2nd éd.; New York, 1911).Google Scholar
5. Feit, W., On the structure of Frobenius groups, Can. J. Math., 9 (1957), 587–596.Google Scholar
6. Gorenstein, D. and Walter, J. H., On finite groups with dihedral Sylow 2-subgroups, Illinois J. Math., 6 (1962), 553–593.Google Scholar
7. Ito, N., On finite groups with given conjugate types I, Nagoya Math. J., 6 (1953), 17–28.Google Scholar
8. Landau, E., Ùber die Klassenzahl der binaren quadratischen Formen von negativer Discriminante, Math. Ann., 56 (1903), 671–676.Google Scholar
9. Miller, G. A., Groups involving only a small number of sets of conjugate operators, Arch. Math. and Phys., 17 (1910), 199–204.Google Scholar
10. Miller, G. A., Groups possessing a small number of sets of conjugate operators, Trans. Amer. Math. Soc, 20 (1919), 260–270.Google Scholar
11. Miller, G. A., Groups involving a small number of sets of conjugate operators, Proc. Nat. Acad. Sci., 20 (1944), 359–362.Google Scholar
12. Poland, J., Two problems on finite groups withk conjugate classes, J. Austral. (Math. Soc. to appear).Google Scholar
13. Schmidt, O. J., Selected works of Otto Schmidt (Moscow, 1959).Google Scholar
14. Scott, W. R., Group Theory (Englewood Cliffs, 1964).Google Scholar
15. Sigley, D. T., Groups involving five complete sets of non-invariant conjugate operators, Duke Math. J., 1 (1935), 477–479.Google Scholar
16. Suzuki, M., A characterization of simple groups LF(2.p), J. Fac. Sci. Univ. Tokyo Sect. I, 6 (1951), 259–293.Google Scholar
17. Suzuki, M., On finite groups containing an element of order four which commutes only with its powers, Illinois J. Math., 8 (1959), 255–271.Google Scholar
18. Zassenhaus, H., Ùber endliche Fastkorper, Hamb. Abh., 11 (1936), 187–220.Google Scholar