Skip to main content Accessibility help
×
Hostname: page-component-7479d7b7d-k7p5g Total loading time: 0 Render date: 2024-07-16T02:45:13.222Z Has data issue: false hasContentIssue false

On Engel and positive laws

Published online by Cambridge University Press:  05 July 2011

O. Macedońska
Affiliation:
Silesian University of Technology
W. Tomaszewski
Affiliation:
Silesian University of Technology
C. M. Campbell
Affiliation:
University of St Andrews, Scotland
M. R. Quick
Affiliation:
University of St Andrews, Scotland
E. F. Robertson
Affiliation:
University of St Andrews, Scotland
C. M. Roney-Dougal
Affiliation:
University of St Andrews, Scotland
G. C. Smith
Affiliation:
University of Bath
G. Traustason
Affiliation:
University of Bath
Get access

Summary

Image of the first page of this content. For PDF version, please use the ‘Save PDF’ preceeding this image.'
Type
Chapter
Information
Publisher: Cambridge University Press
Print publication year: 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] S. I., Adian, The problem of Burnside and identities in groups, Nauka, Moscow, 1975 (Russian), (see also, trans. J., Lennox and J., Wiegold, Ergebnisse der Mathematik und ihrer Grenzgebiete 92, Springer-Verlag, Berlin, 1979).Google Scholar
[2] S., Bachmuth and H. Y., Mochizuki, Third Engel groups and the Macdonald–Neumann conjeture, Bull. Austral. Math. Soc. 5 (1971), 379–386.Google Scholar
[3] R., Baer, Engelsche Elemente Noetherscher Gruppen, Math. Ann. 133 (1957), 256–270.Google Scholar
[4] B., Bajorska, On the smallest locally and residually closed class of groups, containing all finite and all soluble groups, Publ. Math. Debrecen 64 (2006), no. 4, 423–431.Google Scholar
[5] R. G., Burns, O., Macedońska and Yu., Medvedev, Groups satisfying semigroup laws and nilpotent-by-Burnside varieties, J. Algebra 195 (1997), 510–525.Google Scholar
[6] R. G., Burns and Yu., Medvedev, A note on Engel groups and local nilpotence, J. Austral. Math. Soc. (Series A) 64 (1998), 92–100.Google Scholar
[7] R. G., Burns, Yu., Medvedev, Group laws implying virtual nilpotence, J. Austral. Math. Soc. 74 (2003), 295–312.Google Scholar
[8] S. N., Černikov, Infinite nonabelian groups with an invariance condition for infinite nonabelian subgroups, Dokl. Akad. Nauk SSSR 194 (1970), 1280–1283.Google Scholar
[9] G., Endimioni, Bounds for nilpotent-by-finite groups in certain varieties of groups, J. Austral. Math. Soc. 73 (2002), 393–404.Google Scholar
[10] N. D., Gupta, Some group-laws equivalent to the commutative law, Arch. Math. (Basel) 17 (1966), 97–102.Google Scholar
[11] J. R. J., Groves, Varieties of soluble groups and a dichotomy of P.Hall, Bull. Austral. Math. Soc. 5 (1971), 391–410.Google Scholar
[12] K. W., Gruenberg, Two Theorems on Engel Groups, Proc. Cambridge Philos. Soc. 49 (1953), 377–380.Google Scholar
[13] G., Havas and M. R., Vaughan-Lee, 4-Engel groups are locally nilpotent, Internat. J. Algebra and Comput. 15 (2005), no. 4, 649–682.Google Scholar
[14] H., Heineken, Engelsche Elemente der Lange drei, Illinois J. Math. 5 (1961), 681–707.Google Scholar
[15] Y., Kim and A. H., Rhemtulla, On locally graded groups, Proceedings of the Third International Conference on Group Theory, Pusan, Korea 1994 (Springer-Verlag, Berlin—Heidelberg—New York, 1995), 189–197.Google Scholar
[16] Y. K., Kim and A. H., Rhemtulla, Weak maximality condition and polycyclic groups, Proc. Amer. Math. Soc. 123 (1995), 711–714.Google Scholar
[17] F. W., LeviGroups in which the commutator operation satisfies certain algebraic conditions, J. Indian Math. Soc. (N.S.) 6 (1942), 87–97.Google Scholar
[18] J., Lewin and T., LewinSemigroup laws in varieties of soluble groups, Proc. Cambridge Philos. Soc. 65 (1969), 1–9.Google Scholar
[19] Olga, Macedonska, What do the Engel laws and positive laws have in common, Fundamental and Applied Mathematics 14 (2008), no. 7, 175–183 (Russian).Google Scholar
[20] W., Magnus, A., Karrass, D., Solitar, Combinatorial Group Theory, 2nd ed., Dover Publications, New York 1976.Google Scholar
[21] A. I., Mal'tsev, Nilpotent semigroups, Ivanov. Gos. Ped. Inst. Uc. Zap. 4 (1953), 107–111 (Russian).Google Scholar
[22] Yu., MedvedevOn compact Engel groups, Israel J. Math. 135 (2003), no. 1, 147–156.Google Scholar
[23] J., Milnor, Growth of finitely generated solvable groups, J. Diff. Geom. 2 (1968), 447–449.Google Scholar
[24] H., Neumann, Varieties of groups, Springer-Verlag, Berlin, Heidelberg, New York 1967.Google Scholar
[25] A. Yu., Ol'shanskii and A., Storozhev, A group variety defined by a semigroup law, J. Austral. Math. Soc. (Series A) 60 (1996), 255–259.Google Scholar
[26] F., Point, Milnor identities, Comm. Algebra 24 (1996), no. 12, 3725–3744.Google Scholar
[27] B. H., Neumann and T., Taylor, Subsemigroups of nilpotent groups, Proc. Roy. Soc. (Series A) 274 (1963), 1–4.Google Scholar
[28] S., Rosset, A property of groups of non-exponential growth, Proc. Amer. Math. Soc. 54 (1976), 24–26.Google Scholar
[29] J. F., Semple and A., Shalev, Combinatorial conditions in residually finite groups, I, J. Algebra 157 (1993), 43–50.Google Scholar
[30] A. I., Shirshov, On certain near-Engel groups, Algebra i Logika 2 (1963), no. 1, 5–18 (Russian).Google Scholar
[31] G., Traustason, Semigroup identities in 4-Engel groups, J. Group Theory 2 (1999), 39–46.Google Scholar
[32] G., Traustason, A note on the local nilpotence of 4-Engel groups, Internat. J. Algebra Comput. 15 (2005), no. 4, 757–764.Google Scholar
[33] G., Traustason, Milnor groups and (virtual) nilpotence, J. Group Theory 8 (2005), 203–221.Google Scholar
[34] Unsolved problems in group theory: The Kourovka Notebook, Fourteenth edition, Russian Academy of Sciences, Siberian Division, Institute of Mathematics, Novosibirsk 1999.
[35] M., Vaughan-LeeEngel-4 groups of exponent 5, Proc. London Math. Soc. 74 (1997), no. 3, 306–334.Google Scholar
[36] J. S., Wilson, Two-generator conditions for residually finite groups, Bull. London Math. Soc. 23 (1991), 239–248.Google Scholar
[37] J. S., Wilson and E. I., Zelmanov, Identities for Lie algebras of pro-p groups, J. Pure Appl. Algebra 81 (1992), 103–109.Google Scholar

Save book to Kindle

To save this book to your Kindle, first ensure coreplatform@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Available formats
×

Save book to Dropbox

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.

Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

Available formats
×