Hostname: page-component-848d4c4894-v5vhk Total loading time: 0 Render date: 2024-06-19T11:00:20.775Z Has data issue: false hasContentIssue false
  • English
  • Français

The twisted group algebra of a finite nilpotent group over a number field

Published online by Cambridge University Press:  18 May 2009

Hans Opolka
Hans Opolka
Affiliation:
Mathematisches Institut der Universität Münster, Roxelerstrasse 64, 4400 Münster
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let G be a finite group with neutral element e which operates trivially on the multiplicative group R* of a commutative ring with identity 1. Let H2(G, R*) = Z2(G, R*)/B2(G, R*) denote the second cohomology group of G with respect to the trivial G-module R*. With every factor system (2-cocycle) f ∈ Z2(G, R*) we associate the so called (central) twisted group algebra (R, G, f) of G over R (see [4, Chapter V, 23.7] or [13, §4] for a definition). If f is cohomologous to f', then the R-algebras (R, G, f) and (R, G, f′) are isomorphic. Hence, up to R-algebra isomorphism, (R, G, f) is determined by the cohomology class f∈H2(G, R*) determined by f. If R = k is a field of characteristic not dividing the order |G| of G, then a computation of the discriminant of (k, G, f) shows that (k, G, f) is semisimple (see [13, 4.2]).

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 20 , Issue 1 , January 1979 , pp. 55 - 61
Copyright
Copyright © Glasgow Mathematical Journal Trust 1979

References

REFERENCES

1.Barannik, L. F., On the Schur index of projective representations of finite groups, Math. USSR-Sb. 15 (1971), 109–120.Google Scholar
2.Clifford, A. H., Representations induced in an invariant subgroup, Ann. of Math., 38 (1937), 533–550.Google Scholar
3.Fontaine, J.-M., Sur la decomposition des algèbres de groupes, Ann. Sci. Écolé Norm. Sup. (4) 4 (1971), 121–180.Google Scholar
4.Huppert, B., Endliche Gruppen I (Springer, 1967).Google Scholar
5.Janusz, G. J., Some remarks on Clifford's theorem and the Schur index, Pacific J. Math. 32 (1970), 119–129.CrossRefGoogle Scholar
6.Kneser, M., Zum expliziten Reziprozitätsgesetz von I. R. Šafarevič, Math. Nachr. 6 (1951/1952), 89–96.Google Scholar
7.Lenstra, H. W. Jr, K2 of a global field consists of symbols, Algebraic K-theory, Evanston 1976, Lecture Notes in Math. 551 (Springer, 1976).Google Scholar
8.Ng, H. N., Faithful irreducible projective representations of metacyclic groups, J. Algebra 38 (1976), 8–28.CrossRefGoogle Scholar
9.Opolka, H., Rationalitätsfragen bei projektiven Darstellungen endlicher Gruppen, Dissertation, Univ. Münster (1976); to appear in parts in Arch. Math. (Basel) and J. Algebra.Google Scholar
10.Opolka, H., Projective representations of extra-special p-groups, Glasgow Math. J. 19 (1978), 149–152.Google Scholar
11.Reid, A., Twisted group rings which are semiprime Goldie rings, Glasgow Math. J. 16 (1975), 1–11.Google Scholar
12.Roquette, P., Realisierung von Darstellungen endlicher nilpotenter Gruppen, Arch. Math. (Basel) 9 (1958), 241–150.CrossRefGoogle Scholar
13.Yamazaki, K., On projective representations and ring extension of finite groups, J. Fac. Sci. Univ. Tokyo Sect. I 10 (1964), 147–195.Google Scholar
14.Žmud, E. M., Symplectic geometries over finite abelian groups, Math. USSR-Sb. 86 (1971), 9–34.Google Scholar