Nearly isomorphic torsion free abelian groups

P-equivalence de groupes nilpotents

“…La P-6quivalence peut &re caract6ris6e en termes de quotients ou de compl6tion dans la ligne de [8]. Pour cela, supposons P fini, non vide, Si G e t H sont deux groupes nilpotents, il existe par un th6or+me de Blackburn (voir [15], 6.4) u n m dans P tel que G" et H m sont form6s uniquement de puissances pi ..... 3, pour peP.…”

“…(2) pour tout naturel n, il existe q~: G~H avec [-H: Imq~] fini et premier avec n. (groupes <<nearly isomorphic~> -la condition est sym6trique, voir [8]). …”

Conductor of an abelian group

“…If we let R = Z then we have defined when two abelian groups M and N are locally isomorphic. Local isomorphism is referred to as near isomorphism of rtffr groups in [2,12] and as in the same genus class in [14]. We use the one term to define what is essentially one equivalence relationship in our more limited setting.…”

Indecomposable decompositions of torsion-free abelian groups

“…To our knowledge all published examples of groups with "pathological" decompositions are almost completely decomposable groups. The essential simplification occurs by passing from isomorphism to Lady's near-isomorphism ( [9], [10,Chapter 9]). A theorem by David Arnold ([2, Corollary 12.9], [10, Theorem 10.2.5]) says that passing from isomorphism to the coarser equivalence relation of Lady does not mean a loss of generality for our line of questioning: if G and G ′ are nearly isomorphic groups, then P(G) = P(G ′ ).…”