Direct products and elementary equivalence of polycyclic-by-finite groups

JOURNAL OF ALGEBRA

卷:418

Article

Lasserre, C.¹  Oger, F.² 

[1] Inst Fourier UMR 5582, F-38402 St Martin Dheres, France

[2] Univ Paris 07, UFR Math, F-75205 Paris 13, France

关键词: Polycyclic-by-finite groups;    Elementary equivalence;    Direct products;   

DOI  :  10.1016/j.jalgebra.2014.07.006

来源: Elsevier

【 摘 要 】

Generalizing previous results, we give an algebraic characterization of elementary equivalence for polycyclic-by-finite groups. We use this characterization to investigate the relations between their elementary equivalence and the elementary equivalence of the factors in their decompositions in direct products of indecomposable groups. In particular, we prove that the elementary equivalence G equivalent to H of two such groups G, H is equivalent to each of the following properties: (1) G x ... x G (k times G) equivalent to H x ... x H (k times H) for an integer k >= 1; (2) A x G equivalent to B x H for two polycyclic-by-finite groups A, B such that A equivalent to B. It is not presently known if (1) implies G equivalent to H for any groups G, H. (c) 2014 Elsevier Inc. All rights reserved.

【 授权许可】

Free