Abstract
This is an informal report on the present status of the displayed problems in Hanna Neumann’s book Varieties of groups [50]. The reader should have the book at hand, not only for notation and terminology but also because we do not re-state the problems nor repeat the comments available there. Our aim is to he up to date, not to present a complete historical survey; superseded references will be mostly ignored regardless of their significance at the time. Details of solutions will not be quoted from papers already published, unless needed to motivate further questions. The discussion and the problems highlighted in it reflect our personal interests rather than any considered value-judgement.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
С. И. Адян [S. I. Adyan], “Бесконечные неприводимые системы групповых тождеств” [Infinite irreducible systems of group identities], Izv. Akad. Nook SSSR Ser. Mat. 34 (1970), 715–734; Math. USSR-Izv. 4 (1970), 721–739 (1971). MR44#4078.
С. И. Адян [S. I. Adyan], “О подгруппах свободных групп нечетного показателя” [Subgroups of free groups of odd exponent], Trudy Mat. Inst. StekZov. 112 (1971), 64–72.
S.I. Adyan, “Periodic groups of odd exponent”, these Proc.
M.D. Atkinson, “Alternating trilinear forms and groups of exponent 6 ”, J. Austral. Math. Soc. 16 (1973), 111–128.
Seymour Bachmuth, Horace Y. Mochizuki and David Walkup, “A nonsolvable group of exponent 5 ”, Bull. Amer. Math. Soc. 76 (1970), 638–640. MR41#1862.
Gilbert Baumslag, “Some theorems on the free groups of certain product varieties”, J. Combinatorial Theory 2 (1967), 77–79. MR34#5902.
J.M. Brady, “On the classification of just-non-Cross varieties of groups”, Bull. Austral. Math. Soc. 3 (1970), 293–311. MR44#289.
J.M. Brady, R.A. Bryce and John Cossey, “On certain abelian-by-nilpotent varieties”, Bull. Austral. Math. Soc. 1 (1969), 403–416. MR41#1843.
М. А. Бронштейн [M. A. Bronšteĭn], “О Вербальные подгруппы свободных групп” [Verbal subgroups of free groups], Doki. Akad. Nook SSSR 177 (1967), 255–257; Soviet Math. Dokl. 8 (1967), 1386–1388. MR36#3861.
M.S. Brooks, L.G. Kovdcs and M.F. Newman, “A finite basis theorem for product varieties of groups”, Bulll. Austral. Math. Soc. 2 (1970), 39–44. MR42#1882.
Н. Р. Брумберг [N.R. Brumberg], “О вербальных коммутатор двух многообразий групп” [On the commutator of two varieties of groups], Mat. Sb. (NS) 79 (121) (1969), 37–58. MR39#5672.
Roger M. Bryant, “On s-critical groups”, Quart. J. Math. Oxford Ser. (2) 22 (1971), 91–101. MR44#2805.
Roger M. Bryant, “Finite splitting groups in varieties of groups”, Quart. J. Math. Oxford Ser. (2) 22 (1971), 169–172. MR44#5366.
Roger M. Bryant, “On join varieties of groups”, Math. Z. 119 (1971), 143–148. MR44#6293.
Roger M. Bryant, “On locally finite varieties of groups”, Proc. London Math. Soc. (3) 24 (1972), 395–408. MR45#8708.
Roger M. Bryant, “Some infinitely based varieties of groups”, J. Austral. Math. Soc. 16 (1973), 29–32.
R.M. Bryant and M.F. Newman, “Some finitely based varieties of groups”, Proc. London Math. Soc. (3) 28 (1974), 237–252.
R.A. Bryce, “Projective groups in varieties”, Bull. Austral. Math. Soc. 6 (1972), 169–174. MR45#3527.
R.A. Bryce and John Cossey, “Some product varieties of groups”, Bull. Austral. Math. Soc. 3 (1970), 231–264. MR42#4618.
T.C. Chau, “The laws of some nilpotent groups of small rank” (PhD thesis, Australian National University, Canberra, 1968; Abstract: Bull. Austral. Math. Soc. 2 (1970), 277–279 ).
T.G. Chau, “The laws of some nilpotent groups of small rank”, J. Austral. Math. Soc. (to appear).
D.E. Cohen, “On the laws of a metabelian variety”, J. Algebra 5 (1967), 267–273. MR44#5929.
John Cossey, “On a problem of Hanna Neumann”, Math. Z. 106 (1968), 187–190. MR37#6357.
John Cossey, “Some classes of indecomposable varieties of groups”, J. Austral. Math. Soc. 9 (1969), 387–398. MR40#4341.
John Cossey, “On decomposable varieties of groups”, J. Austral. Math. Soc. 11 (1970), 340–342. MR42#3155.
M.J. Dunwoody, “On product varieties”, Math. Z. 104 (1968), 91–97. MR37#291.
J.R.J. Groves, “On varieties of soluble groups”, Bull. Austral. Math. Soc. 5 (1971), 95–109. MR45#3528.
J.R.J. Groves, “Varieties of soluble groups and a dichotomy of P. Hall”, Bull. Austral. Math. Soc. 5 (1971), 391–410. Zb1. 217, 76.
J.R.J. Groves, “On varieties of soluble groups II”, Bull. Austral. Math. Soc. 7 (1972), 437–441. Zbl.241.20023.
J.R.J. Groves, “An extension of a dichotomy of P. Hall to some varieties of groups”, Arch. der Math. 23 (1972), 573–580.
J.R.J. Groves, “On some finiteness conditions for varieties of metanilpotent groups”, Arch. der Math. 24 (1973), 252–268.
Chantier Kanta Gupta, “The free centre-by-metabelian groups”, J. Austral. Math. Soc. 16 (1973), 294–299.
Narain Gupta and Frank Levin, “Generating groups of certain soluble varieties”, J. Austral. Math. Soc. (to appear).
Hermann Heineken and Peter M. Neumann, “Identical relations and decision procedures for groups”, J. Austral. Math. Soc. 7 (1967), 39–47. MR34#5931.
G.H. Houghton, “Direct decomposability of reduced free groups”, J. London Math. Sac. 43 (1968), 534–538. MR37#2840.
C.H. Houghton, “Directly decomposable finite relatively free groups”, J. London Math. Soc. (2) 4 (1971), 381–384. MR46#7382.
Gareth A. Jones, “Varieties and simple groups”, J. Austral. Math. Soc. (to appear).
Bjarni Jonsson, “The sum of two finitely based lattice varieties need not be finitely based”, Notices Amer. Math. Soc. 21 (1974), A-2.
Bjarni Jonsson, “Sums of finitely based lattice varieties”, preprint.
М. И. Каргаполов, В. А. Чуркин [M.I. Kargapolov, V.A. Curkin], “О многообразиях разрешимых групп” [On varieties of soluble groups], Algebra i Logika 10 (1971), 651–657.
Otto H. Kegel and Bertram A.F. Wehrfritz, Locally finite groups (North-Holland Mathematical Library, 3. North-Holland, Amsterdam, London; American Elsevier, New York, 1973 ).
Ю. Г. Клейман [Yu.G. Kleĭman], “О базисах продукта многообразий групп”, [On bases for product varieties of groups], Lay. Akad. Nauk SSSR Ser. Mat. 37 (1973), 95–97.
L.C. Kovács, “On the number of varieties of groups”, J. Austral. Math. Soc. 8 (1968), 444–446. MR37#5277.
L.G. Kovács, “A remark on critical groups”, J. Austral. Math. Soc. 9 (1969), 465–466. MR39#5674.
L.G. Kovács, “Varieties and finite groups”, J. Austral,. Math. Soc. 10 (1969), 5–19. MR40#1459.
L.G. Kovács, “Inaccessible varieties of groups”, J. Austral. Math. Soc. (to appear).
L.G. Kovács, M.F. Newman and P.F. Pentony, “Generating groups of nilpotent varieties”, Bull. Amer. Math. Soc. 74 (1968), 968–971. MR37#5276.
L.G. Kovâcs and M.R. Vaughan-Lee, “A problem of Hanna Neumann on closed sets of group words”, Bull. Austral. Math. Soc. 5 (1971), 341–342. MR45#3533.
Frank Levin, “Generating groups for nilpotent varieties”, J. Austral. Math. Soc. 11 (1970), 28–32. MR41#1844.
Stephen Meskin, “Some varieties without the amalgam embedding properties”, Bull. Austral. Math. Soc. 1 (1969), 417–418. MR41#1845.
Hanna Neumann, Varieties of groups (Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 37. Springer-Verlag, Berlin, Heidelberg, New York, 1967). MR35#6734.
Peter M. Neumann, “Splitting groups and projectives in varieties of groups”, Quart. J. Math. Oxford Ser. (2) 18 (1967), 325–332. MR36#3859.
Peter M. Neumann, “A note on the direct decomposability of relatively free groups”, Quart. J. Math. Oxford Ser. (2) 19 (1968), 67–79. MR36#6485.
M.F. Newman, “Just non-finitely-based varieties of groups”, Bull. Austral,. Math. Soc. 4 (1971), 343–348. MR43#4891.
А.Ю. Ольшанский [A.Ju. Ol’šanskiĭ], “Об одной задаче Ханны Нейман” [On a problem of Hanna Neumann], Mat. Sb. (NS) 76 (118) (1968), 449–453. MR37#2841.
А.Ю. Ольшанский [A.Ju. Ol’šanskiĭ], “О На проблеме конечного базиса тождеств групп” [On the problem of a finite basis for the identities of groups], Izv. Akad. Nauk SSSR 34 (1970), 376–384; Math. USSR-Izv. 4 (1970), 381–389 (1971). MR44#4079.
А.Ю. Ольшанский [A.Ju, Ol’šanskiĭ], “Растворимые просто кроссовых многообразий групп” [Soluble just non-Cross varieties of groups], Mat. Sb. (NS) 85 (127) (1971), 115–131; Math. USSR-Sb. 14 (1971), 115–129 (1972).
Paul Pentony, “Laws in torsion-free nilpotent varieties with particular reference to the laws of free nilpotent groups” (PhD thesis, Australian National University, Canberra, 1970; Abstract: Bull. Austral. Math. Soc. 5 (1971), 283–284).
Ю.П. Размыслов [Ju.P. Razmyslov], “Об ангела алгебры Ли” [On Engel Lie algebras], Algebra i Logika 10 (1971), 33–44; Algebra and Logic 10 (1971), 21–29. MR45#3498.
Ю.П. Размыслов [Ju.P. Razmyslov], “Об Примером нерастворимого просто кроссовы различных групп” [An example of an insoluble just-non-Cross variety of groups], Algebra i Logiker 11 (1972), 186–205.
A.G.R. Stewart, “On centre-extended-by-metabelian groups” (PhD thesis, Australian National University, Canberra, 1968 ).
A.G.R. Stewart, “On centre-extended-by-metabelian groups”, Math. Ann. 185 (1970), 285–302. MR41#6945.
M.R. Vaughan-Lee, “Uncountably many varieties of groups”, Bull. London Math. Soc. 2 (1970), 280–286. MR43#2054.
M.R. Vaughan-Lee, “Generating groups of nilpotent varieties”, Bull. Austral. Math. Soc. 3 (1970), 145–154. MR43#2091.
M.R. Vaughan-Lee, “On product varieties of groups”, Bull. Austral. Math. Soc. 5 (1971), 239–240. MR45#3529.
M.A. Ward, “Basic commutators”, Philos. Trans. Roy. Soc. London Ser. A 264 (1969), 343–412. MR40#4379.
James Joseph Woeppel, “Finite groups generating a product variety” (PhD thesis, University of Illinois at Urbana-Champaign, 1970 ).
James J. Woeppel, “Join-irreducible Cross product varieties of groups”, Trans. Amer. Math. Soc. (to appear).
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1974 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Kovács, L.G., Newman, M.F. (1974). Hanna Neumann’s Problems on Varieties of Groups. In: Newman, M.F. (eds) Proceedings of the Second International Conference on the Theory of Groups. Lecture Notes in Mathematics, vol 372. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-21571-5_42
Download citation
DOI: https://doi.org/10.1007/978-3-662-21571-5_42
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-06845-7
Online ISBN: 978-3-662-21571-5
eBook Packages: Springer Book Archive