On the direct product of 𝑉-groups
| Autor: | Donald P. Minassian |
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| Rok vydání: | 1971 |
| Předmět: | |
| Zdroj: | Proceedings of the American Mathematical Society. 30:434-436 |
| ISSN: | 1088-6826 0002-9939 |
| DOI: | 10.1090/s0002-9939-1971-0286727-5 |
| Popis: | Let G and H be ordered groups such that every full order on a subgroup extends to a full order on the group; then the direct product, GXH, need not have this property. In fact a stronger result holds. A group is called a V-group if every full order on a subgroup may be extended to a full order on the group. Kargapolov [2 ] showed that an arbitrary torsion-free group G is a V-group if and only if G has an abelian normal subgroup B such that (1) the factor group G/B is abelian, and (2) for arbitrary : in B and a in G-B there are positive integers m 5 n such that alt3ma = fn.l In particular, torsion-free abelian groups are V-groups. The necessity of Kargapolov's theorem is essentially due to Terehov [6 ]. LEMMA. Let G have the presentation G = { a, b I ba = abP }, where p is a fixed prime. Then G is a V-group. PROOF. First of all, G is torsion-free since it admits a full order [ ] (direct verification is not difficult). Next, we construct a subgroup B of G satisfying the conditions of Kargapolov's theorem. By definition of G, bP = a-lba, and so by induction bpk = a-kbak, all k > 0. Moreover (akba-k)Pk = ak(bk) a-k = b, so that we may define bp-k =akbak, all k > 0. |
| Databáze: | OpenAIRE |
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