|
In the mathematical field of group theory, the Kurosh subgroup theorem describes the algebraic structure of subgroups of free products of groups. The theorem was obtained by Alexander Kurosh, a Russian mathematician, in 1934.〔A. G. Kurosh, ''Die Untergruppen der freien Produkte von beliebigen Gruppen.'' Mathematische Annalen, vol. 109 (1934), pp. 647–660.〕 Informally, the theorem says that every subgroup of a free product is itself a free product of a free group and of its intersections with the conjugates of the factors of the original free product. ==History and generalizations== After the original 1934 proof of Kurosh, there were many subsequent proofs of the Kurosh subgroup theorem, including proofs of Kuhn (1952),〔H. W. Kuhn. ''Subgroup theorems for groups presented by generators and relations.'' Annals of Mathematics (2), vol. 56, (1952), pp. 22–46〕 Mac Lane (1958)〔S. Mac Lane. ''A proof of the subgroup theorem for free products.'' Mathematika, vol. 5 (1958), pp. 13–19〕 and others. The theorem was also generalized for describing subgroups of amalgamated free products and HNN extensions.〔A. Karrass, and D. Solitar. ''The subgroups of a free product of two groups with an amalgamated subgroup.'' Transactions of the American Mathematical Society, vol. 150 (1970), pp. 227–255.〕〔A. Karrass, and D. Solitar. ''Subgroups of HNN groups and groups with one defining relation''. Canadian Journal of Mathematics, vol. 23 (1971), pp. 627–643.〕 Other generalizations include considering subgroups of free pro-finite products and a version of the Kurosh subgroup theorem for topological groups.〔P. Nickolas. ''A Kurosh subgroup theorem for topological groups.'' Proceedings of the London Mathematical Society (3), vol. 42 (1981), no. 3, pp. 461–477〕 In modern terms, the Kurosh subgroup theorem is a straightforward corollary of the basic structural results of Bass–Serre theory about groups acting on trees.〔Daniel Cohen. ''Combinatorial group theory: a topological approach.'' London Mathematical Society Student Texts, 14. Cambridge University Press, Cambridge, 1989. ISBN 0-521-34133-7; 0-521-34936-2〕 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Kurosh subgroup theorem」の詳細全文を読む スポンサード リンク
|