  Transcriber's note: In the fall of 1976, my advisor, David Mumford, handed me
a short preprint by George Kempf to read. It was the first state of what
eventually became his influential Annals paper "Instability in Invariant
Theory" (Annals of Mathematics, Second Series, Vol. 108, No. 2 (Sep., 1978),
pp. 299-316). The introduction to the published version ends with an
acknowledgement and a dig: "I want to thank the referee of the first version of
this paper for pointing out Corollary 4-5 and conjecturing that the original
{0}-instability could be replaced by S-instability. Unfortunately, the
inclusion of S-instability has completely destroyed the simplicity of the
original version." Over the intervening years, the simplicity and elegance of
the first version has continued to create a readership for it, and copies (see
Figure 8) continue to circulate informally. I created this LaTeX'ed version to
make the paper accessible to all who may be interested in it, trying to keep
the look close to that of the original typewritten preprint and making changes
only to correct a few obvious typos and harmonize the markup. My thanks to
George's children, Robin and Lucas Kempf, for graciously granting me permission
to post this transcription. Comments welcome to Ian Morrison
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