On a theorem of J. A. Green
Brauer's famous induction theorem states that each irreducible complex character of a finite group \(G\) is expressible as an integral linear combination of linear characters induced from elementary subgroups. (An elementary subgroup is a direct product of a \(p\)-subgroup with a cyclic \(p'\)-subgroup, for some prime \(p\).) \textit{J. A. Green} [Proc. Camb. Philos. Soc. 51, 237-239 (1954; Zbl 0057.02102)] showed conversely that if \(\mathcal H\) is a family of subgroups of \(G\) with the property that each irreducible character of \(G\) is expressible as an integral linear combination of characters induced from subgroups in \(\mathcal H\), then each elementary subgroup of \(G\) is contained in a conjugate of some subgroup in \(\mathcal H\). The author of the present paper offers a new proof of Green's theorem, without using the formula for induced characters. The new ingredient of his proof is the use of the characteristic class functions of \(G\), that is, the functions that take the value 1 on all elements of a given conjugacy class, and 0 on all other elements. Examination of Section 15 F of \textit{C. W. Curtis} and \textit{I. Reiner}, Methods of Representation Theory, Volume 1 [John Wiley, New York (1981; Zbl 0469.20001)], suggests that the author's proof differs little in its details from existing proofs, and retains the usual final contradiction, that a certain prime divides the value of the principal character on some element.
linear combinations of linear characters
scientific article; zbMATH DE number 1247623
scientific article