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- W2062138621 abstract "This paper continues a study of the character ring of a finite group begun in [S]. In [S] we characterized generalized permutation characters-that is, integral linear combinations of permutation characters of a group G. Our characterization involved, for each prime p dividing 1 G ] and each cyclic p’subgroup (x) of G, p-Sylow subgroups P, of C,(x) and pX of N&x) chosen so that P, E P,. To determine if a generalized character x of G is a generalized permutation character, one checks whether for each p and (x) as above a certain generalized character xX of P, extends to a suitable p-integral combination of characters of i),. In this paper we will characterize integral linear combinations of characters AC where A is a linear character of a subgroup of G and the order of A divides a fixed integer m. We will call such characters IZG m-monomial characters. Integral linear combinations of m-monomial characters are called generalized m-monomial characters. We may assume, of course, that m divides ( G I. When m= 1, an m-monomial character is precisely a transitive permutation character, integral linear combinations of which were characterized in ]5] as described above. When m = ] G ], the m-monomial characters include all monomial characters of G, and by Brauer’ s induction theorem, any generalized character of G is generalized m-monomial. Thus, the two extreme cases m = 1 and m = ] G 1 have been dealt with, so the purpose of this paper is to establish intermediate theorems between the main theorem of [S] and Braucr’s characterization of characters. Since we work mostly with p-integral rather than integral linear combinations of characters the main theorem of this paper in the case m = ] G 1 will yield not exactly Brauer’s characterization of characters but rather a p-integral version, due to J. Thompson, which says that a class function x on G is a 123" @default.
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- W2062138621 date "1981-07-01" @default.
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- W2062138621 title "A characterization of generalized m-monomial characters" @default.
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- W2062138621 doi "https://doi.org/10.1016/0021-8693(81)90109-5" @default.
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