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On the Character Rings of Finite Groups

Published online by Cambridge University Press:  20 November 2018

B. Banaschewski
B. Banaschewski*
Affiliation:
McMaster University, Hamilton, Ontario
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The characters of the representations of a finite group G over a field K of characteristic zero generate a ring oK(G) of functions on G, the K-character ring of G, which is readily seen to be Zϕ1 + . . . + Zϕn, where Z is the ring of rational integers and ϕ1, . . . , ϕn are the characters of the different irreducible representations of G over K. The theorem that every irreducible representation of G over an algebraically closed field Ω of characteristic zero is equivalent to a representation of G over the subfield of Ω which is generated by the g0th roots of unity (g0 the exponent of G) was proved by Brauer (4) via the theorems that

(1) OΩ(G) is additively generated by the induced characters of representations of elementary subgroups of G, and

(2) the irreducible representations over 12 of any elementary group are induced by one-dimensional subgroup representations (3).

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 15 , 1963 , pp. 605 - 612
Copyright
Copyright © Canadian Mathematical Society 1963

References

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