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On the number of conjugacy classes of normalisers in a finite p-group

Published online by Cambridge University Press:  17 April 2009

Norberto Gavioli ,
Leire Legarreta ,
Carmela Sica  and
Maria Tota
Norberto Gavioli
Affiliation:
Dipartimento di Matematica Pura ed Applicata, Università di L'Aquila, Via Vetoio, 67010 Coppito (L'Aquila), Italy e-mail: gavioli@univaq.it
Leire Legarreta
Affiliation:
Dipartimento di Matematica e Informatica, Università di Salerno, Via Ponte don Melillo, 84084 Fisciano (SA), Italy e-mail: csica@unisa.it
Carmela Sica
Affiliation:
Matematika Saila, Euskal Herriko Unibertsitatea, 48080 Bilbo, Spain e-mail: mtplesol@lg.ehu.es
Maria Tota
Affiliation:
Dipartimento di Matematica e Informatica, Università di Salerno, Via Ponte don Melillo, 84084 Fisciano (SA), Italy e-mail: mtota@unisa.it
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In 1996 Poland and Rhemtulla proved that the number v (G) of conjugacy classes of non-normal subgroups of a non-Hamiltonian nilpotent group G is at least c − 1, where c is the nilpotency class of G. In this paper we consider the map that associates to every conjugacy class of subgroups of a finite p-group the conjugacy class of the normaliser of any of its representatives. In spite of the fact that this map need not be injective, we prove that, for p odd, the number of conjugacy classes of normalisers in a finite p-group is at least c (taking into account the normaliser of the normal subgroups). In the case of p-groups of maximal class we can find a better lower bound that depends also on the prime p.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 73 , Issue 2 , April 2006 , pp. 219 - 230
Copyright
Copyright © Australian Mathematical Society 2006

References

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