The Greatest Mathematical Paper of All Time

You will say that my title is absurd. "Mathematical papers cannot be totally ordered. It's a great pity! Poor old Coleman has obviously gone berserk in his old age." Please read on. If in 1940 you had asked the starry-eyed Canadian graduate student who was lapping up the K-calculus from Alonzo Church in Princeton to name the single most important mathematics paper, without doubt I would have chosen Kurt G6del's bombshell [12] that had rocked the foundat ions of mathematics a few years before. In 1970, after my twenty years of refereeing and reviewing, if you had posed the same question, without any hesitation I would have chosen the enormous paper of Walter Feit and John Thompson [11] confirming Burnside's 1911 conjecture [3] that simple finite groups have even order. Now, in the autumnal serenity of semi-retirement, having finally looked at some of Wilhelm Killing's writings, without any doubt or hesitation I choose his paper dated "Braunsberg, 2 Februar, 1888" as the most significant mathematical paper I have read or heard about in fifty years. Few can contest my choice since apart from Engel, Umlauf, Molien, and Cartan few seem to have read it. Even my friend Hans Zassenhaus, whose Liesche Ringe (1940) was a landmark in the subject, admitted over our second beer at the American Mathematical Society meeting in January 1987 that he had not read a word of Killing. Presupposing that my reader has a rudimentary understanding of linear algebra and group theory, I shall attempt to explain the main new ideas introduced in Killing's paper, describe its remarkable results, and suggest some of its subsequent effects. The paper that,