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The minimal degree of a faithful quasi-permutation representation of an abelian group

Published online by Cambridge University Press:  18 May 2009

Houshang Behravesh
Houshang Behravesh
Affiliation:
Department of Mathematics, University of Manchester, Manchester M13 9PL, England
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Let G be a finite linear group of degree n; that is, a finite group of automorphisms of an n-dimensional complex vector space (or, equivalently, a finite group of non-singular matrices of order n with complex coefficients). We shall say that G is a quasi-permutation group if the trace of every element of G is a non-negative rational integer. The reason for this terminology is that, if G is a permutation group of degree n, its elements, considered as acting on the elements of a basis of an n -dimensional complex vector space V, induce automorphisms of V forming a group isomorphic to G. The trace of the automorphism corresponding to an element x of G is equal to the number of letters left fixed by x, and so is a non-negative integer. Thus, a permutation group of degree n has a representation as a quasi-permutation group of degree n. See [5].

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 39 , Issue 1 , January 1997 , pp. 51 - 57
Copyright
Copyright © Glasgow Mathematical Journal Trust 1997

References

REFERENCES

1.Burns, J. M., Goldsmith, B., Hartley, B. and Sandling, R., On quasi-permutation representations of finite groups, Glasgow Math. J. 36 (1994), 301–308.CrossRefGoogle Scholar
2.Hoffman, M., An invariant of finite abelian groups. Amer. Math. Monthly 94 (1987), 664–666.CrossRefGoogle Scholar
3.Isaacs, I. M., Character theory of finite groups (Academic Press, New York, 1976).Google Scholar
4.Sehgal, S. K., Topics in group rings (Marcel Dekker, New York, 1978).Google Scholar
5.Wong, W. J., Linear groups analogous to permutation groups, J. Austral. Math. Soc. Ser. A 3 (1963), 180–184.CrossRefGoogle Scholar
6.Wong, W. J., On linear p-groups, J. Austral. Math. Soc. Ser. A 4 (1964), 174–178.CrossRefGoogle Scholar