196 research outputs found

    A Characterization ofp-Radical Groups

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    Elementary Trigonometric Sums related to Quadratic Residues

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    Let p be a prime = 3 (mod 4). A number of elegant number-theoretical properties of the sums T(p) = \sqrt{p}sum_{n=1}^{(p-1)/2} tan(n^2\pi/p) and C(p) = \sqrt{p}sum_{n=1}^{(p-1)/2} cot(n^2\pi/p) are proved. For example, T(p) equals p times the excess of the odd quadratic residues over the even ones in the set {1,2,...,p-1}; this number is positive if p = 3 (mod 8) and negative if p = 7 (mod 8). In this revised version the connection of these sums with the class-number h(-p) is also discussed. For example, a very simple formula expressing h(-p) by means of the aforementioned excess is proved. The bibliography has been considerably enriched. This article is of an expository nature.Comment: A number of misprints have been corrected and one or two improvements have been done to the previous version of the paper with same title. The paper will appear to Elem. der Mat

    Brauer characters and the Harris–Knörr correspondence in p-solvable groups

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    AbstractIf b is a p-block of a normal subgroup N of a p-solvable group G and b* is its Brauer correspondent in NN(L), where L is a defect group of b, then Harris–Knörr correspondents over b and b* contain equal numbers of irreducible Brauer characters over height zero Brauer characters

    Relative π-Blocks of π-Separable Groups

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    Polynomial solutions of differential equations

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