Hostname: page-component-586b7cd67f-2brh9 Total loading time: 0 Render date: 2024-11-26T05:42:51.275Z Has data issue: false hasContentIssue false
  • English
  • Français

Verbal wreath products and certain product varieties of groups

Published online by Cambridge University Press:  09 April 2009

R. G. Burns
R. G. Burns
Affiliation:
Monash University Melbourne
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Recently A. L. Šmel'kin [14] proved that a product variety1 is generated by a finite group if and only if is nilpotent, is abelian, and the exponents of and are coprime. Alternatively, by the theorem of Oates and Powell [13], we may say that a Cross variety is decomposable if and only if it is of the above form.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 7 , Issue 3 , August 1967 , pp. 356 - 374
Copyright
Copyright © Australian Mathematical Society 1967

References

[1]Baumslag, Gilbert, Neumann, B. H, Neumann, Hanna and Neumann, Peter M., ‘On varieties generated by a finitely generated group’. Math. Z. 86 (1964), 93–122.CrossRefGoogle Scholar
[2]Burns, R. G., Ph. D. thesis, Australian National Univeristy, 1966.Google Scholar
[3]Dunwoody, M. J., ‘On verbal subgroups of free groups’. Arch. Math., 16 (1965), 153–157.CrossRefGoogle Scholar
[4]Gaschütz, Wolfgang, ‘Über die Φ-Untergruppe endlicher Gruppen’. Math. Z., 58 (1953), 160–170.CrossRefGoogle Scholar
[5]Hall, Marshall Jr,The theory of groups. (New York 1959).Google Scholar
[6]Higman, Graham. ‘Some remarks on varieties of groups’. Quarterly J. Math., Oxford (2) 10 (1959), 165–178.CrossRefGoogle Scholar
[7]Kovács, L. G. and Newman, M. F.. ‘Cross varieties of groups’. Submitted to Proc. Roy. Soc.Google Scholar
[8]Kurosh, A. G.. The theory of groups, Vols. 1 and 11. (New York 1956).Google Scholar
[9]Moran, S.. Associative operations on groups I. Proc. London Math. Soc. (3) 6 (1956), 581–596.CrossRefGoogle Scholar
[10]Moran, S., ‘The homomorphic image of the intersection of a verbal subgroup and the cartesian subgroup of a free product’. J. London. Math. Soc., 33 (1958), 237–245.CrossRefGoogle Scholar
[11]Neumann, B. H.. ‘Identical relations in groups I’. Math. Ann., 114 (1937), 506–525.CrossRefGoogle Scholar
[12]Peter, M. Neumann. ‘On the structure of standard wreath products of groups’. Math. Z., 84 (1964) 343–373.Google Scholar
[13]Oates, Sheila and Powell, M. B., ‘Identical relations in finite groups’. J. Algebra, I (1964), 11–39.CrossRefGoogle Scholar
[14]Šel'kin, A. L.. ‘Wreath products and varieties of groups’. Izv. Nauk S.S.S.R. Ser. Mat., 29 (1965), 149–170. (Also for summary in English: Soviet Mathematics. Vol. 5. No. 4 (1964). Translation of Dokl. Akad. Nauk S.S.S.R. for Am. Math. Soc).Google Scholar
[15]Weichsel, Paul M., ‘On critical p-groups’. Proc. London Math. Soc. 14 (1964), 83–100.CrossRefGoogle Scholar