Springer Singapore Acta Medica Okayama 0251-4184 40 3 2015 Classification of the Linearly Reductive Finite Subgroup Schemes of SL2 527 534 EN Mitsuyasu Hashimoto Department of Mathematics, Okayama University We classify the linearly reductive finite subgroup schemes G of SL2=SL(V) over an algebraically closed field k of positive characteristic, up to conjugation. As a corollary, we prove that such G is in one-to-one correspondence with an isomorphism class of two-dimensional F-rational Gorenstein complete local rings with the coefficient field k by the correspondence G↦((SymV)G) ˆ. No potential conflict of interest relevant to this article was reported.