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Concerning the general equations of the seventh and eighth degrees. (English) JFM 29.0080.01
Die Arbeit behandelt die Beziehungen der allgemeinen Gleichung achten Grades \(F_8(x)=0\) nach Adjunction der Quadratwurzel aus ihrer Discriminante und ihrer Totalresolvente \(15^{\text{ten}}\) Grades \(G_{15}(y)=0\); ebenso die Relationen der allgemeinen Gleichung siebenten Grades nach Adjunction der Quadratwurzel aus ihrer Discriminante und ihrer Totalresolvente \(15^{\text{ten}}\) Grades \(H(z)=0\). Im Anschluss namentlich an Noether (Math. Ann. 16) und Jordan (Traité des Substitutions, No. 426, 516) studirt der Verf. die linearen Tripelsysteme \(\Delta_{2^k-1}\) bei \(2^k-1\) Elementen und die ihnen entsprechende Substitutionengruppe \(G_{(2^k-1)(2^k-2^1)\dots(2^k-2^{k-1})}^{2^k-1}\) sowie die 30 Quadrupelsysteme \(\square_8\) für acht Elemente und die entsprechenden 30 Substitutionengruppen \(G_{8.168}^8\) und beweist, ähnlich wie Jordan, dass die alternirende Gruppe \(G_{\frac12.8!}^8\) und die lineare homogene Gruppe \(LHG_{\frac12.8!}^{15}\) holoedrisch isomorph sind. Von diesem letzteren Satze wird dann noch ein rein gruppentheoretischer Beweis gegeben.
Den Schluss bildet die Verwendung der vorhergehenden Entwickelungen für die Lösung des Kirkman’schen Problems der 15 Schulmädchen.

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[1] There is contact here with my papers:Concerning Jordan’s Linear Groups (Bulletin of the American Mathematical Society, vol. 2, pp. 33-43, 1895),Tactical Memoranda I?III (American Journal of Mathematics, vol. 18, 264-303, 1896).
[2] It is to be noted that in introducing the notion of the group of isomorphisms of a group (loc. cit.,Concerning Jordan’s Linear Groups (Bulletin of the American Mathematical Society, vol. 2, pp. 223, 1895), and especially in applying it in connection with the Abelian group of orderp n and type (111... ton units) to the study of the linear groups (loc. cit.:Concerning Jordan’s Linear Groups (Bulletin of the American Mathematical Society, vol. 2, pp. 336-339, 342, 1895) Burnside should have cited my papers:The Group of Holoedric Transformation into Itself of a given Group, andConcerning Jordan’s Linear Groups (Bulletin of the American Mathematical Society, vol. 1, pp. 61-66, 1894, and vol. 2, pp. 33-43, 1895). Cf. the foot-note near the close of §1 of the present paper.
[3] This well-known doubling process first occurs in a paper by Horner:On Triads of Once-Paired Elements (Quarterly Journal of Mathematics, vol. 9, pp. 15-18, 1868). He speaks however explicitly only of triple systems which may be arranged ?in ascending order? (loc. cit.,Concerning Jordan’s Linear Groups (Bulletin of the American Mathematical Society, vol. 2, p. 15, 1895).
[4] Moore:Concerning the Abstract Group... (loc. cit.,Concerning Jordan’s Linear Groups (Bulletin of the American Mathematical Society, vol. 2, pp 33-43, 1895, theorems B and C). · JFM 26.0173.02
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