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Deformation Methods and the Strong Unbounded Representation Type of p-Groups

Published online by Cambridge University Press:  22 January 2016

J. D. Donald  and
F. J. Flanigan
J. D. Donald
Affiliation:
Birchwood, Tyringham, Mass, Department of Mathematics, San Diego State University
F. J. Flanigan
Affiliation:
Birchwood, Tyringham, Mass, Department of Mathematics, San Diego State University
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A basic problem in the representation theory of a finite group G is the determination of all indecomposable G-modules. Thus, for G = C(n) = a cyclic group of order n over an arbitrary field, the indecomposable representations, finite in number, are known from the theory of a single linear transformation. In 1954 Higman [9] showed that, in sharp contrast to the classical case of characteristic zero, an arbitrary finite group G has indecomposables of arbitrarily high dimension over any field of prime characteristic p iff the p-Sylow subgroup of G is non-cyclic (cf. unbounded representation type [3, p. 431]). Examples published by Heller and Reiner [8] in 1961 indicated that this phenomenon is even more extensive; reinterpreting a result of Dieudonné [4] as classifying the indecomposable modules for a square zero algebra on two generators, they showed that G = C(p) × C(p) (and therefore many other groups) has infinitely many non-isomorphic indecomposables in every even dimension over an infinite field of characteristic p (cf. strong unbounded representation type).

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 57 , June 1975 , pp. 107 - 119
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1975

References

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