Geodesic
The twisted group algebra of a finite nilpotent group over a number field
Glasgow mathematical journal, Tome 20 (1979) no. 1, pp. 55-61

Voir la notice de l'article provenant de la source Cambridge University Press

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]). 
Opolka, Hans. The twisted group algebra of a finite nilpotent group over a number field. Glasgow mathematical journal, Tome 20 (1979) no. 1, pp. 55-61. doi: 10.1017/S0017089500003724
@article{10_1017_S0017089500003724,
     author = {Opolka, Hans},
     title = {The twisted group algebra of a finite nilpotent group over a number field},
     journal = {Glasgow mathematical journal},
     pages = {55--61},
     year = {1979},
     volume = {20},
     number = {1},
     doi = {10.1017/S0017089500003724},
     url = {http://geodesic.mathdoc.fr/articles/10.1017/S0017089500003724/}
}
TY  - JOUR
AU  - Opolka, Hans
TI  - The twisted group algebra of a finite nilpotent group over a number field
JO  - Glasgow mathematical journal
PY  - 1979
SP  - 55
EP  - 61
VL  - 20
IS  - 1
UR  - http://geodesic.mathdoc.fr/articles/10.1017/S0017089500003724/
DO  - 10.1017/S0017089500003724
ID  - 10_1017_S0017089500003724
ER  - 
%0 Journal Article
%A Opolka, Hans
%T The twisted group algebra of a finite nilpotent group over a number field
%J Glasgow mathematical journal
%D 1979
%P 55-61
%V 20
%N 1
%U http://geodesic.mathdoc.fr/articles/10.1017/S0017089500003724/
%R 10.1017/S0017089500003724
%F 10_1017_S0017089500003724

[1] 1.Barannik, L. F., On the Schur index of projective representations of finite groups, Math. USSR-Sb. 15 (1971), 109–120. Google Scholar

[2] 2.Clifford, A. H., Representations induced in an invariant subgroup, Ann. of Math., 38 (1937), 533–550. Google Scholar

[3] 3.Fontaine, J.-M., Sur la decomposition des algèbres de groupes, Ann. Sci. Écolé Norm. Sup. (4) 4 (1971), 121–180. Google Scholar

[4] 4.Huppert, B., Endliche Gruppen I (Springer, 1967). Google Scholar

[5] 5.Janusz, G. J., Some remarks on Clifford's theorem and the Schur index, Pacific J. Math. 32 (1970), 119–129. Google Scholar | DOI

[6] 6.Kneser, M., Zum expliziten Reziprozitätsgesetz von I. R. Šafarevič, Math. Nachr. 6 (1951/1952), 89–96. Google Scholar

[7] 7.Lenstra, H. W. Jr, K of a global field consists of symbols, Algebraic K-theory, Evanston 1976, Lecture Notes in Math. 551 (Springer, 1976). Google Scholar

[8] 8.Ng, H. N., Faithful irreducible projective representations of metacyclic groups, J. Algebra 38 (1976), 8–28. Google Scholar | DOI

[9] 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] 10.Opolka, H., Projective representations of extra-special p-groups, Glasgow Math. J. 19 (1978), 149–152. Google Scholar

[11] 11.Reid, A., Twisted group rings which are semiprime Goldie rings, Glasgow Math. J. 16 (1975), 1–11. Google Scholar

[12] 12.Roquette, P., Realisierung von Darstellungen endlicher nilpotenter Gruppen, Arch. Math. (Basel) 9 (1958), 241–150. Google Scholar | DOI

[13] 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] 14.Žmud, E. M., Symplectic geometries over finite abelian groups, Math. USSR-Sb. 86 (1971), 9–34. Google Scholar

Cité par Sources :