On varietal capability of Infinite direct products of groups

Document Type: Research Paper


Department of Pure Mathematics, Center of Excellence in Analysis on Algebraic Structures, Ferdowsi University of Mashhad, P. O. Box 1159-91775, Mashhad, Iran


Recently‎, ‎the authors gave some conditions under which a direct product‎ ‎of finitely many groups is $\mathcal{V}-$capable if and only if each of its‎ ‎factors is $\mathcal{V}-$capable for some varieties $\mathcal{V}$‎. ‎In this paper‎, ‎we extend this fact to any infinite direct product of groups‎. ‎Moreover‎, ‎we conclude some results for $\mathcal{V}-$capability of direct products of infinitely many groups in varieties of abelian‎, ‎nilpotent and polynilpotent groups‎.


Main Subjects

R. Baer (1938). Groups with preassigned central and central quotient groups. Trans. Amer. Math. Soc.. (44), 378-412
F. R. Beyl, U. Felgner, P. Schmid (1979). On groups occurring as center factor groups. J. Algebra. (61), 161-177
G. Ellis (1998). On groups with a finite nilpotent upper central quotient. Arch. Math.. (70), 89-96
P. Hall (1940). The classification of prime-power groups. J. reine angew. Math.. (182), 130-141
M. Hall, Jr., J. K. Senior (1964). The Groups of Order $2^n (nleq 6)$. Macmillan, New York.
G. Karpilovsky (1987). The Schur Multiplier. London Math. Soc. Monographs, New Series 2, Clarendon Press, Oxford University Press, Oxford.
H. Mirebrahimi, B. Mashayekhy On varietal capability of direct product of groups. J. Adv. Res. Pure Math., to appear..
M. R. R. Moghaddam (1980). The Baer-invarient and the direct limit. Monatsh. Math.. (90), 37-43
M. R. R. Moghaddam (1980). The Baer-invarient of a direct product. Arch. Math.. (33), 504-511
M. R. R. Moghaddam, S. Keyvanfar (1997). A new notion derived from varieties of groups. Algebra Colloq.. (4:1), 1-11
D. J. S. Robinson (1996). A Course in the Theory of Groups. Second Edition, Springer-Verlag.