Timeline for Every abelian torsion-free group is strictly totally orderable (via the compactness theorem)
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| Jun 15, 2020 at 7:27 | history | edited | CommunityBot |
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| Feb 16, 2020 at 18:33 | comment | added | YCor | Another note for record: a group $G$ for which $\mu'_1(G)=\infty$ (i.e., for every $g\neq 1$ the semigroup generated by the conjugacy class of $g$ does not contain $1$) is called an $R^*$-group by Fuchs (On Orderable Groups, in Proc. Internat. Conf. Theory of Groups, Austral. Nat. Univ., Canberra, 89-98 (1965)) and such groups are considered in N. Gupta, A. Rhemtulla, On ordered groups. Algebra Univ. 1 (1971/72), 129-132. | |
| Feb 16, 2020 at 17:53 | comment | added | YCor | Note: Rhemtulla (Proc AMS 1973) proved that a group that is residually-$p$ for infinitely many primes $p$ is bi-orderable. The above proposition is a generalization of this. Indeed, every group that is residually (finite with no element of prime order $\le n$) for every $n$, embeds into a torsion-free ultraproduct of finite groups. The argument is in spirit the same, and actually Rhemtulla provides a quantitative bound for $\mu'_k(C_p)=\mu_k(C_p)$. | |
| Feb 16, 2020 at 0:11 | history | answered | YCor | CC BY-SA 4.0 |