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The Reidemeister-Schreier and Kuroš subgroup theorems

  • A. J. Weir (a1)

If a group G is presented in terms of generators and relations, then the classical Reidemeister-Schreier Theorem [1] gives a presentation for any subgroup of G. If G is a free product of groups Gα each of which is presented in terms of generators and relations, then the main result of this paper is a presentation for any subgroup H of G, which shows the nature of H as a free product of certain subgroups of G. This result is a generalization of the celebrated Kuroš Theorem [2]. It also includes the Reidemeister–Schreier Theorem and the Schreier Theorem [1] which states that any subgroup of a free group is free.

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1.Sohreier O., Abh. Math. Sem. Hamburg Univ., 5 (1927), 161183.
2.Kurosoh A., Math. Ann., 109 (1934), 647660.
3.Baer R. and Levi F., Compositio Math., 3 (1936), 391398.
4.Takahasi M., Proc. Imp. Acad. Tokyo, 20 (1944), 589594.
5.Hall M., Pacific Journal of Math., 3 (1953), 115120.
6.Kuhn H. W., Annals of Math., 56 (1952), 2246.
7.Fox R. H., Annals of Math., 57 (1953), 547560; 59 (1954), 196–210.
8.Kuroš A., Theory of Groups (Chelsea, New York, 1956), vol. 2, §33.
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  • ISSN: 0025-5793
  • EISSN: 2041-7942
  • URL: /core/journals/mathematika
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