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Injective modules for group algebras of locally finite groups

Published online by Cambridge University Press:  24 October 2008

I. M. Musson
I. M. Musson
Affiliation:
University of Warwick
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Extract

Two recent results relate the existence of injective modules for group algebras which are ‘small’ in some sense to the structure of the group.

(1) The trivial kG-module is injective if and only if G is a locally finite group with no elements of order p = char k (9).

(2) If (G) is a countable group, then every irreducible kG-module is injective if and only if G is a locally finite p′ group which is abelian-by-finite (9) and (11)

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 84 , Issue 2 , September 1978 , pp. 247 - 262
Copyright
Copyright © Cambridge Philosophical Society 1978

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References

REFERENCES

(1)Artin, E., Nesbitt, C. and Thrall, R. M.Rings with minimum, condition (Ann Arbor; University of Michigan Press, 1943).Google Scholar
(2)Bourbaki, N.Algebre, chapitre 8 (Paris: Hermann, 1958).Google Scholar
(3)Brauer, R. and Feit, W.An analogue of Jordan's theorem in characteristic p. Ann. of Math. 84 (1966), 119–131.CrossRefGoogle Scholar
(4)Burgess, W.Rings of quotients of group rings. Canad. J. Math. 21 (1969), 865–875.CrossRefGoogle Scholar
(5)Cailleau, A.Une caractérisation des modules ∑-injectifs. C. R. Acad. Sci. Paris, Ser. A-B, 269 (1969), 997–999.Google Scholar
(6)Curtis, C. W. and Reiner, I. Representation theory of finite groups and associative algebras. Pure and applied mathematics, vol. IX (New York, London: Interscience, John Wiley and Sons, 1962).Google Scholar
(7)Faith, C.Rings with ascending condition on annihilators. Nagoya Math. J. 27 (1966), 179–191.CrossRefGoogle Scholar
(8)Faith, C.Modules finite over endomorphism ring. Lectures on rings and modules. Lecture Notes in Mathematics, no. 246 (1972).CrossRefGoogle Scholar
(9)Farkas, D. R. and Snider, R. L.Group algebras whose simple modules are injective. Trans. Amer. Math. Soc. 194 (1974), 241–248.CrossRefGoogle Scholar
(10)Hartley, B. and McDougall, D.Injective modules and soluble groups satisfying the minimum condition on normal subgroups. Bull. Austral. Math. Soc. 4 (1971), 113–135.CrossRefGoogle Scholar
(11)Hartley, B.Injective modules over group rings. Quart. J. Math. Oxford Ser. 28 (1977), 1–29.CrossRefGoogle Scholar
(12)Kegel, O. H. and Wehrfritz, B. A. F.Locally finite groups (Amsterdam: North Holland, 1973).Google Scholar
(13)Lambek, J.Lectures on rings and modules (Ginn–Blaisdell, 1966).Google Scholar
(14)Lawrence, J. A countable self-injective ring is quasi-Frobenius. (To appear.)Google Scholar
(15)Passman, D. S.Infinite group rings (New York: Marcel Dekker, 1971).Google Scholar
(16)Passman, D. S.The algebraic structure of group rings (Wiley: Interscience, 1977).Google Scholar
(17)Renault, G.Sur les anneaux des groupes. C. R. Acad. Sci. Paris, Ser. A–B 273 (1971), 84–87.Google Scholar