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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

On classical Clifford theory


Author: Morton E. Harris
Journal: Trans. Amer. Math. Soc. 309 (1988), 831-842
MSC: Primary 20C20; Secondary 16A26, 20C05
MathSciNet review: 961616
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Abstract: Let $ k$ be a field, let $ N$ be a normal subgroup of a finite group $ H$ and let $ M$ be a completely reducible $ k[N]$-module. We give sufficient conditions for a finite dimensional (finite) group crossed product $ k$-algebra to be a Frobenius or symmetric $ k$-algebra. These results imply that $ k[H]/(J(k[N])k[H])$ and the endomorphism $ k$-algebra, $ {\operatorname{End} _{k[H]}}({M^H})$, of the induced module $ {M^H}$ are symmetric $ k$-algebras. We also completely describe the $ k[H]$-indecomposable decomposition of $ {M^H}$. It follows that the head and socle of an indecomposable component of $ {M^H}$ are irreducible isomorphic $ k[H]$-modules.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1988-0961616-6
PII: S 0002-9947(1988)0961616-6
Article copyright: © Copyright 1988 American Mathematical Society