zbMATH — the first resource for mathematics

Uncountable groups have many nonconjugate subgroups. (English) Zbl 0629.20001
The whole of this substantial paper is devoted to the proof of the theorem: Main theorem. If G is a group of cardinality \(\lambda\), \(\lambda\) an uncountable cardinal, and \(\mu =Min\{\mu:\) \(2^{\mu}\geq \lambda \}\), then \(nc_{\leq \mu}(G)\geq \lambda\). (Where \(nc_{\leq \mu}(G)\) is the number of pairwise nonconjugate subgroups of G of power \(\mu\).) This theorem has the interesting conclusion: Conclusion. If \(\lambda\) is an uncountable cardinal and G a group of cardinality \(\lambda\) then G has at least \(\lambda\) pairwise nonconjugate subgroups of power less than \(\lambda\). This paper completes the work begun by the author in a previous paper [Algebra Univers. 16, 131-146 (1983; Zbl 0521.20015)] where the result was proved under GCH and for many values of \(\lambda\) (for every G). The techniques used are those of mathematical logic, and the author thoughtfully provides an appendix for nonlogicians containing the facts from mathematical logic that are required for an understanding of the paper.
Reviewer: Sh.Oates-Williams

20A15 Applications of logic to group theory
20E07 Subgroup theorems; subgroup growth
03E55 Large cardinals
Full Text: DOI
[1] Chang, C.C.; Keisler, H.J., Model theory, (1973), North-Holland Amsterdam · Zbl 0276.02032
[2] Devlin, K.; Shelah, S., A weak form of the diamond which follows from \(2\^{}\{ℵ0\} < 2\^{}\{ℵ1\}\), Israel J. math., 29, 239-247, (1978)
[3] Engelkingand, R.; Karlowicz, M., Some theorems of set theory and their topological consequences, Fund. math., 57, 275-285, (1965) · Zbl 0137.41904
[4] E. Rips, to appear.
[5] Shelah, S., On the number of non-conjugate subgroups, Algebra universalis, 16, 131-146, (1983) · Zbl 0521.20015
[6] Shelah, S., A problem of kurosh, Jonsson groups and applications, (), 373-394
[7] Shelah, S., Classification theory, (1978), North-Holland Amsterdam · Zbl 0388.03009
[8] S. Shelah, Abstracts Amer. Math. Soc.
[9] Shelah, S., Proper forcing, () · Zbl 0495.03035
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.