Abstract
We shall prove that every group of cardinality ℵ1 has at least ℵ1 non conjugate subgroups, and we shall generalize this theorem to many more uncountable cardinalities. For example underGCH for every uncountable cardinal λ and every groupG of cardinality λ,G has at least λ non conjugate subgroups.
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References
SShelah,A problem of Kurosh, Jonsson groups and applications. Proc. of Symp. in Oxford, July, 1976.Word problem II. ed Adjan, Boone and Higman, North Holland Publ. Co. 1979.
S.Shelah,Classification theory. North Holland Publ. Co. 1978.
E.Rips,Generalized small cancellation theory, submitted to Israel Journal of Mathematics.
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I would like to thank Rami Grossberg for writing and rewriting this paper, and Wilfrid Hodges for removing many errors and suggesting improvements in presentation; many facts are proved only due to his explicit request.
This research was supported by grant (No. 1110) from the United States-Israel Binational Science Foundation.
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Shelah, S. On the number of non conjugate subgroups. Algebra Universalis 16, 131–146 (1983). https://doi.org/10.1007/BF01191760
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DOI: https://doi.org/10.1007/BF01191760