An explicit formula for the Picard group of the cyclic group of order

Author:
Alexander Stolin

Journal:
Proc. Amer. Math. Soc. **121** (1994), 375-383

MSC:
Primary 11R65; Secondary 11R21, 19A31

DOI:
https://doi.org/10.1090/S0002-9939-1994-1243832-X

MathSciNet review:
1243832

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Abstract | References | Similar Articles | Additional Information

Abstract: We give a formula for the Picard group of the integer group ring of the cyclic group of order for any odd prime *p*. As a corollary one gets a formula for properly irregular prime *p* in terms of Bernoulli numbers.

**[B]**H. Bass,*Algebraic K-theory*, Benjamin, New York, 1968. MR**0249491 (40:2736)****[B-Sh]**Z. I. Borevich and I. R. Shafarevich,*Number theory*, Academic Press, New York, 1966. MR**0195803 (33:4001)****[C-F]**J. W. S. Cassels and A. Fröhlich,*Algebraic number theory*, Thompson Book Co., Washington, DC, 1967. MR**0215665 (35:6500)****[C-R]**C. W. Curtis and I. Reiner,*Methods of representation theory*, vol. II, Wiley, New York, 1987. MR**892316 (88f:20002)****[G]**S. Galovich,*The class group of a cyclic p-group*, J. Algebra**30**(1974), 368-387. MR**0476838 (57:16390a)****[K-M]**M. A. Kervaire and M. P. Murthy,*On the projective class group of cyclic groups of prime power order*, Comment. Math. Helv.**52**(1977), 415-452. MR**0476693 (57:16252)****[M]**J. Milnor,*Introduction to algebraic K-theory*, Ann. of Math. Stud., vol. 72, Princeton Univ. Press, Princeton, NJ, 1971. MR**0349811 (50:2304)****[S1]**A. Stolin,*On the*-*group of the integer group ring of the cyclic group of order*, Proceeding of the All-Union Conference on the Theory of Rings, Algebras, Modules, Kishinev, 1980. (Russian)**[S2]**-,*On the*-*group of the integer group ring of the cyclic group of order*, preprint, Kharkov University, 1980. (Russian)**[S3]**-,*On the Picard group of the integer group ring of a cyclic p-group and of rings close to them*, preprint, Kharkov Univ., 1984. (Russian)**[U]**S. Ullom,*Fine structure of class groups of cyclic p-groups*, J. Algebra**49**(1977), 112-124. MR**0491601 (58:10823)**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1994-1243832-X

Article copyright:
© Copyright 1994
American Mathematical Society