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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
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Abstract

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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

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