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On projective characters of prime degree

Published online by Cambridge University Press:  18 May 2009

R. J. Higgs
R. J. Higgs
Affiliation:
Department of Mathematics, University College, Belfield, Dublin 4, Ireland.
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All groups G considered in this paper are finite and all representations of G are defined over the field of complex numbers. The reader unfamiliar with projective representations is referred to [9] for basic definitions and elementary results.

Let Proj(G, α) denote the set of irreducible projective characters of a group G with cocycle α. In a previous paper [3] the author showed that if G is a (p, α)-group, that is the degrees of the elements of Proj(G, α) are all powers of a prime number p, then G is solvable. However Isaacs and Passman in [8] were able to give structural information about a group G for which ξ(1) divides pe for all ξ ∈ Proj(G, 1), where 1 denotes the trivial cocycle of G, and indeed classified all such groups in the case e = l. Their results rely on the fact that G has a normal abelian p-complement, which is false in general if G is a (p, α)-group; the alternating group A4 providing an easy counter-example for p = 2.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 33 , Issue 3 , September 1991 , pp. 311 - 321
Copyright
Copyright © Glasgow Mathematical Journal Trust 1991

References

REFERENCES

1.DeMeyer, F. R. and Janusz, G. J., Finite groups with an irreducible representation of large degree. Math. Z. 108 (1969), 145–153.CrossRefGoogle Scholar
2.Higgs, R. J., On the degrees of projective representations. Glasgow Math. J. 30 (1988), 133–135.CrossRefGoogle Scholar
3.Higgs, R. J., Groups whose projective character degrees are powers of a prime. Glasgow Math. J. 30 (1988), 177–180.CrossRefGoogle Scholar
4.Higgs, R. J., Projective characters of degree one and the inflation-restriction sequence. J. Austral. Math Soc. (Series A) 46 (1989), 272–280.CrossRefGoogle Scholar
5.Hoffman, P. N. and Humphreys, J. F., Non-isomorphic groups with the same projective character tables. Comm. Algebra 15 (1987), 1637–1648.CrossRefGoogle Scholar
6.Huppert, B., Endliche Gruppen. I. Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen 134, (Springer-Verlag, Berlin, Heidelberg, New York, 1967).CrossRefGoogle Scholar
7.Isaacs, I. M., Character theory of finite groups. (Pure and Applied Mathematics, a series of monographs and textbooks; 69), (Academic Press, New York, London, 1976).Google Scholar
8.Isaacs, I. M. and Passman, D. S., Groups whose irreducible representations have degrees dividing pe. Illinois J. Math. 8 (1964), 446–457.CrossRefGoogle Scholar
9.Karpilovsky, G., Projective representations of finite groups. (Monographs and Textbooks in Pure and Applied Mathematics; 94), (Marcel Dekker, New York, Basel, 1985).Google Scholar
10.Karpilovsky, G., The Schur multiplier. (London Mathematical Society monographs; new ser., 2), (Oxford University Press, Oxford, New York, 1987).Google Scholar
11.Ng, H. N., Degrees of irreducible projective representations of finite groups. J. London Math. Soc. (2) 10 (1975), 379–384.CrossRefGoogle Scholar
12.Read, E. W., On the centre of a representation group. J. London Math. Soc. (2) 16 (1977), 43–50.CrossRefGoogle Scholar
13.Tahara, K. I., On the second cohomology groups of semidirect products. Math Z. 129 (1972), 365–379.CrossRefGoogle Scholar
14.Thomas, A. D. and Wood, G. V., Group tables. (Shiva mathematics series; 2), (Shiva, Orpington, 1980).Google Scholar