Hostname: page-component-78c5997874-dh8gc Total loading time: 0 Render date: 2024-11-16T18:23:18.344Z Has data issue: false hasContentIssue false
  • English
  • Français

Some bounds for the degree of commutativity of a p-group of maximal class

Published online by Cambridge University Press:  17 April 2009

Antonio Vera-López  and
Gustavo A. Fernández-Alcober
Antonio Vera-López
Affiliation:
Departamento de Matemáticas, Universidad del País Vasco, Bilbao, Spain, e-mail: [email protected]@lg.ehu.es
Gustavo A. Fernández-Alcober
Affiliation:
Departamento de Matemáticas, Universidad del País Vasco, Bilbao, Spain, e-mail: [email protected]@lg.ehu.es
Rights & Permissions [Opens in a new window]

Abstract

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.

In this paper we obtain several lower bounds for the degree of commutativity of a p-group of maximal class of order pm. All the bounds known up to now involve the prime p and are almost useless for small m. We introduce a new invariant b which is related with the commutator structure of the group G and get a bound depending only on b and m, not on p. As a consequence, we bound the derived length of G and the nilpotency class of a certain maximal subgroup in terms of b. On the other hand, we also generalise some results of Blackburn. Examples are given in order to check the sharpness of the bounds.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 51 , Issue 3 , June 1995 , pp. 353 - 367
Copyright
Copyright © Australian Mathematical Society 1995

References

[1]Blackburn, N., ‘On a special class of p-groups’, Acta Math. 100 (1958), 45–92.CrossRefGoogle Scholar
[2]Huppert, B., Endliche Gruppen I (Springer-Verlag, Berlin, Heidelberg, New York, 1967).CrossRefGoogle Scholar
[3]Kovács, L.G. and Leedham-Green, C.R., ‘Some normally monomial p-groups of maximal class and large derived length’, Quart. J. Math. Oxford Ser. (2) 37 (1986), 49–54.CrossRefGoogle Scholar
[4]Lazard, M., ‘Sur les groupes nilpotents et les anneaux de Lie’, Ann. Sci. École Norm. Sup. (3) 71 (1954), 101–190.CrossRefGoogle Scholar
[5]Leedham-Green, C.R. and McKay, S., ‘On p-groups of maximal class, I’, Quart. J. Math. Oxford Ser. (2) 27 (1976), 297–311.CrossRefGoogle Scholar
[6]Leedham-Green, C.R. and McKay, S., ‘On p-groups of maximal class, III’, Quart. J. Math. Oxford Ser. (2) 29 (1978), 281–299.CrossRefGoogle Scholar
[7]Miech, R.J., ‘Metabelian p-groups of maximal class’, Trans. Amer. Math. Soc. 152 (1970), 331–373.Google Scholar
[8]Shepherd, R., p-groups of maximal class, Ph.D. Thesis (University of Chicago, 1970).Google Scholar
[9]Vera-López, A. and Fernández-Alcober, G.A., ‘On p-groups of maximal class, III’, Math. Proc. Cambridge Philos. Soc. 109 (1991), 489–507.CrossRefGoogle Scholar
[10]Vera-López, A. and Fernández-Alcober, G.A., ‘The conjugacy vector of a p-group of maximal class’, Israel J. Math. 86 (1994), 233–252.CrossRefGoogle Scholar
[11]Vera-López, A. and Larrea, B., ‘On p-groups of maximal class’, J. Algebra 137 (1991), 77–116.CrossRefGoogle Scholar
[12]Zassenhaus, H., The theory of groups (Chelsea, New York, 1958).Google Scholar