Théorème de raffinement de Schreier

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En mathématiques, et plus particulièrement en théorie des groupes, le théorème de raffinement de Schreier dit que pour deux suites de composition d'un même groupe, il existe toujours un raffinement de la première et un raffinement de la seconde qui sont équivalents[1]. (Par suite de composition d'un groupe G, on entend ici une suite finie décroissante de sous-groupes de G allant de G à {1}, chacun de ces sous-groupes, à partir du second, étant sous-groupe normal du précédent.)

Ce théorème est nommé d'après le mathématicien autrichien Otto Schreier, qui le démontra en 1928[2]. Il fournit une démonstration du théorème de Jordan-Hölder.

Notes et références

(en) Cet article est partiellement ou en totalité issu de l’article de Wikipédia en anglais intitulé « Schreier refinement theorem » (voir la liste des auteurs).
  1. ↑ Pour une démonstration, voir par exemple N. Bourbaki, Algèbre I, Chapitres 1 à 3, Paris, 1970, p. I.40, ou encore J. Calais, Éléments de théorie des groupes, Paris, 1984, p. 227-229, ou encore S. Lang, Algèbre, 3e édition révisée, Paris, 2004, p. 24.
  2. ↑ Otto Schreier, « Über den Jordan-Hölderschen Satz », Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, vol. 6, 1928, p. 300. Référence donnée par H. J. Zassenhaus, The Theory of Groups, 2e éd., 1958, réimpr. Dover 1999, p. 256.

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