Abstract
There is a continuum of 3-generator soluble non-Hopfian groups that generate pairwise distinct varieties of groups. Each countable (soluble) group is subnormally embeddable into a 3-generator (soluble) non-Hopfian group. As an illustration to a problem of Neumann, we find a continuum of nonmetanilpotent varieties that contain finitely generated non-Hopfian groups and contain uncountably many pairwise nonisomorphic finitely generated groups.
This is a preview of subscription content, log in via an institution to check access.
References
R. Dark, “On subnormal embedding theorems of groups,” J. London Math. Soc., 43, 387–390 (1968).
J. R. J. Groves, “On some finiteness conditions for varieties of metanilpotent groups,” Arch. Math., 24, 252–268 (1973).
P. Hall, “Finiteness conditions for soluble groups,” Proc. London Math. Soc. (3), 4, 419–436 (1954).
P. Hall, “The Frattini subgroups of finitely generated groups,” Proc. London Math. Soc. (3), 11, 327–352 (1961).
H. Heineken, “Normal embeddings of p-groups into p-groups,” Proc. Edinburgh Math. Soc., 35, 309–314 (1992).
H. Heineken, “On normal embedding of subgroups,” Geom. Dedicata, 83, 211–216 (2000).
H. Heineken and V. H. Mikaelian, “On normal verbal embeddings of groups,” J. Math. Sci., 100, No. 1, 1915–1924 (2000).
G. Higman, “A finitely related group with an isomorphic proper factor group,” J. London Math. Soc., 26, 59–61 (1951).
H. Hopf, “Beiträge zur Klassifizierung der Flächenabbildungen,” J. Reine Angew. Math., 165, 225–236 (1931).
R. C. Lyndon and P. E. Schupp, Combinatorial Group Theory, Springer, Berlin (1977).
A. I. Mal’cev, “On isomorphic matrix representations of infinite groups,” Mat. Sb., 8, 405–422 (1940).
C. F. Miller and P. E. Schupp, “On embeddings into Hopfian groups,” J. Algebra, 17, 171–176 (1971).
V. H. Mikaelian, “Subnormal embedding theorems for groups,” J. London Math. Soc., 62, 398–406 (2000).
V. H. Mikaelian, “On embeddings of countable generalized soluble groups in two-generated groups,” J. Algebra, 250, 1–17 (2002).
V. H. Mikaelian, “An embedding construction for ordered groups,” J. Aust. Math. Soc. (A), 74, 379–392 (2003).
V. H. Mikaelian, “Infinitely many not locally soluble SI*-groups,” Ricerche Mat., 52, 1–19 (2003).
V. H. Mikaelian, “On embedding properties of SD-groups,” Int. J. Math. Math. Sci., No. 2, 65–76 (2004).
B. H. Neumann, “A two-generator group isomorphic to a proper factor group,” J. London Math. Soc., 25, 247–248 (1950).
B. H. Neumann, “On a problem of Hopf,” J. London Math. Soc., 28, 351–353 (1953).
B. H. Neumann, “An essay on free products of groups with amalgamations,” Philos. Trans. Roy. Soc. London Ser. A, 246, 503–554 (1954).
B. H. Neumann, “Embedding theorems for ordered groups,” J. London Math. Soc., 35, 503–512 (1960).
B. H. Neumann, “Embedding theorems for groups,” Nieuw Arch. Wisk. (3), 16, 73–78 (1968).
B. H. Neumann and H. Neumann, “Embedding theorems for groups,” J. London Math. Soc., 34, 465–479 (1959).
H. Neumann, Varieties of Groups, Springer, Berlin (1967).
A. Yu. Ol’shanskii, “On the problem of finite base of identities in groups,” Izv. Akad. Nauk SSSR, Ser. Mat. 34, No. 2, 376–384 (1970).
D. J. S. Robinson, A Course in the Theory of Groups, Springer, Berlin (1996).
P. E. Schupp, “A note on non-Hopfian groups,” J. London Math. Soc. (2), 16, 235–236 (1977).
Author information
Consortia
Additional information
Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 15, No. 1, pp. 81–98, 2009.
Rights and permissions
About this article
Cite this article
V. H. Mikaelian. On finitely generated soluble non-Hopfian groups. J Math Sci 166, 743–755 (2010). https://doi.org/10.1007/s10958-010-9890-4
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10958-010-9890-4