Skip to main content Accessibility help
×
Hostname: page-component-848d4c4894-4rdrl Total loading time: 0 Render date: 2024-06-20T05:45:23.886Z Has data issue: false hasContentIssue false

Engel elements in groups

Published online by Cambridge University Press:  05 July 2011

Alireza Abdollahi
Affiliation:
University of Isfahan, Iran
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] A., Abdollahi, Left 3-Engel elements in groups, J. Pure Appl. Algebra 188 (2004), 1–6.Google Scholar
[2] A., Abdollahi and H., Khosravi, When right n-Engel elements of a group form a subgroup? (http://arxiv.org/abs/0906.2439v1).
[3] A., Abdollahi and H., Khosravi, On the right and left 4-Engel elements, to appear in Comm. Algebra (http://arxiv.org/abs/0903.691v2).
[4] S. I., Adjan and P. S., Novikov, Commutative subgroups and the conjugacy problem in free periodic groups of odd order, Izv. Akad. Nauk. SSSR Ser. Mat. 32 (1968), 1176–1190.Google Scholar
[5] R., Baer, Nil-Gruppen, Math. Z. 62 (1955), 402–437.Google Scholar
[6] R., Baer, Engelsche Elemente Noetherscher Gruppen, Math. Ann. 133 (1957), 256–270.Google Scholar
[7] V., Bludov, An example of not Engel group generated by Engel elements, in abstracts of A Conference in Honor of Adalbert Bovdi's 70th Birthday, November 18–23, 2005.Google Scholar
[8] C. J. B., Brookes, Engel elements of soluble groups, Bull. London Math. Soc. 18 (1986), 7–10.Google Scholar
[9] L. V., Dolbak, On the Engel length of the product of Engel elements, Sibirsk. Mat. Zh. 47 (2006), no. 1, 69–72.Google Scholar
[10] H., Fitting, Beiträge zur Theorie der Gruppen endlicher Ordnung, Jahreber. Deutsch. Math. Verein. 48 (1938), 77–141.Google Scholar
[11] ,The GAP Group, GAP–Groups, Algorithms and Programming, version 4.4, available at http://www.gap-system.org, 2005.
[12] E. S., Golod, On nil-algebras and finitely approximable p-groups, Izv. Akad. Nauk SSSR Ser. Mat. 28 (1964), 273–276.Google Scholar
[13] R. I., Grigorchuk, On Burnside's problem on periodical groups, Functional Analysis and Applications 14 (1980), no. 1, 53–54.Google Scholar
[14] K. W., Gruenberg, The Engel elements of a soluble group, Illinois J. Math. 3 (1959), 151–169.Google Scholar
[15] K. W., Gruenberg, The upper central series in soluble groups, Illinois J. Math. 5 (1961), 436–66.Google Scholar
[16] K. W., Gruenberg, The Engel structure of linear groups, J. Algebra 3 (1966), 291–303.Google Scholar
[17] N. D., Gupta and F., Levin, On soluble Engel groups and Lie algebra, Arch. Math. (Basel) 34 (1980), 289–295.Google Scholar
[18] G., Havas and M. R., Vaughan-Lee, 4-Engel groups are locally nilpotent, Internat. J. Algebra Comput. 15 (2005), 649–682.Google Scholar
[19] H., Heineken, Eine Bemerkung Über engelsche Elemente, Arch. Math. (Basel) 11 (1960), 321.Google Scholar
[20] H., Heineken, Engelsche Elemente der Länge drei, Illinois J. Math. 5 (1961), 681–707.Google Scholar
[21] D., Held, On bounded Engel elements in groups, J. Algebra 3 (1966), 360–365.Google Scholar
[22] D., Hilbert, Mathematical Problems, Bull. Amer. Math. Soc. 8 (1902), 437–479.Google Scholar
[23] K. A., Hirsch, Über lokal-nilpotente Gruppen, Math. Z. 63 (1955), 290–294.Google Scholar
[24] S. V., Ivanov, The free Burnside groups of sufficiently large exponents, Internat. J. Algebra Comput. 4 (1994), no. 1–2, ii+308 pp.Google Scholar
[25] S. V., Ivanov and A., Yu. Ol'shanskii, On finite and locally finited subgroups of free Burnside groups of large even exponents, J. Algebra 195 (1997), 241–284.Google Scholar
[26] L.-C., Kappe, Right and left Engel elements in groups, Comm. Algebra 9 (1981), 1295–1306.Google Scholar
[27] L. C., Kappe and P. M., Ratchford, On centralizer-like subgroups associated with the n-Engel word, Algebra Colloq. 6 (1999), 1–8.Google Scholar
[28] W. P., Kappe, Die A-Norm einer Gruppe, Illinois J. Math. 5 (1961), 187–197.Google Scholar
[29] Y. K., Kim and A. H., Rhemtulla, Weak maximality condition and polycyclic groups, Proc. Amer. Math. Soc. 123 (1995), 711–714.Google Scholar
[30] ,The Kourovka Notebook. Unsolved Problems in Group Theory. Sixteenth edition. Edited by V. D., Mazurov and E. I., Khukhro. Russian Academy of Sciences Siberian Division, Institute of Mathematics, Novosibirsk, 2006.Google Scholar
[31] F. W., Levi, Groups in which the commutator operation satisfies certain algebraic conditions, J. Indian Math. Soc. 6 (1942), 87–97.Google Scholar
[32] I. G., Lysenok, Infinite Burnside groups of even exponent, Izv. Math. 60 (1960), no. 3, 453–654.Google Scholar
[33] I. D., Macdonald, Some examples in the theory of groups, in Mathematical Essays dedicated to A. J. Macintyre, (Ohio University Press, Athens, Ohio, 1970), 263–269.Google Scholar
[34] J. E., Martin and J. A., Pamphilon, Engel elements in groups with the minimal condition, J. London Math. Soc. (2) 6 (1973) 281–285.Google Scholar
[35] Martin, L. Newell, On right-Engel elements of length three, Proc. Roy. Irish Acad. Sect. A 96 (1996), no. 1, 17–24.Google Scholar
[36] M. F., Newman and W., Nickel, Engel elements in groups, J. Pure Appl. Algebra 96 (1994), no. 1, 39–45.Google Scholar
[37] W., Nickel, NQ, 1998, A refereed GAP 4 package, see [11].
[38] W., Nickel, Some groups with right Engel elements, in Groups St Andrews 1997, Vol. 2, London Math. Soc. Lecture Note Ser. 261 (CUP, Cambridge, 1999), 571–578.Google Scholar
[39] T. A., Peng, Engel elements of groups with maximal condition on abelian subgroups, Nanta Math. 1 (1966), 23–28.Google Scholar
[40] B. I., Plotkin, On some criteria of locally nilpotent groups, Uspehi Mat. Nauk. 9 (1954), 181–186.Google Scholar
[41] B. I., Plotkin, Radical groups, Mat. Sb. 37 (1955), 507–526.Google Scholar
[42] B. I., Plotkin, Radicals and nil-elements in groups, Izv. Vysš Učebn. Zaved. Matematika 1 (1958), 130–138.Google Scholar
[43] Derek, J. S. Robinson, Finiteness Conditions and Generalized Soluble Groups, Parts I, II, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 62 (Springer-Verlag, New York 1972).Google Scholar
[44] Derek, J. S. Robinson, A Course in the Theory of Groups, Second Edition, Graduate Texts Math. 80 (Springer-Verlag, New York 1996).Google Scholar
[45] E., Schenkman, A generalization of the central elements of a group, Pacific J. Math. 3 (1953), 501–504.Google Scholar
[46] A., Shalev, Combinatorial conditions in Residually finite groups, II, J. Algebra 157 (1993), 51–62.Google Scholar
[47] G., Traustason, On 4-Engel groups, J. Algebra 178 (1995), 414–429.Google Scholar
[48] G., Traustason, Locally nilpotent 4-Engel groups are Fitting groups, J. Algebra 270 (2003), 7–27.Google Scholar
[49] M., Vaughan-Lee, On 4-Engel groups, LMS J. Comput. Math. 10 (2007), 341–353.Google Scholar
[50] Frank, O. Wagner, Nilpotency in groups with the minimal condition on centralizers, J. Algebra 217 (1999), no. 2, 448–460.Google Scholar
[51] B. A. F., Wehrfritz, Groups of automorphisms of soluble groups, Proc. London Math. Soc. (3) 20 (1970), 101–122.Google Scholar
[52] B. A. F., Wehrfritz, Nilpotence in finitary skew linear groups, J. Pure Appl. Algebra 83 (1992), 27–41.Google Scholar
[53] B. A. F., Wehrfritz, Finitary automorphism groups over commutative rings, J. Pure Appl. Algebra 172 (2002) 337–346.Google Scholar
[54] E. I., Zel'manov, The solution of the restricted Burnside problem for groups of odd exponent, Math. USSR Izvestia 36 (1991), 41–60.Google Scholar
[55] E. I., Zel'manov, The solution of the restricted Burnside problem for 2-groups, Math. Sb. 182 (1991), 568–592.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
×