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ITÔ’S THEOREM ON GROUPS WITH TWO CLASS SIZES REVISITED

Part of: Structure and classification of infinite or finite groups

Published online by Cambridge University Press:  19 October 2011

ELENA ALEMANY ,
ANTONIO BELTRÁN  and
MARÍA JOSÉ FELIPE
ELENA ALEMANY*
Affiliation:
Departamento de Matemática Aplicada, Universidad Politécnica de Valencia, 46022 Valencia, Spain (email: ealemany@mat.upv.es)
ANTONIO BELTRÁN
Affiliation:
Departamento de Matemáticas, Universidad Jaume I, 12071 Castellón, Spain (email: abeltran@mat.uji.es)
MARÍA JOSÉ FELIPE
Affiliation:
Instituto Universitario de Matemática Pura y Aplicada, Universidad Politécnica de Valencia, 46022 Valencia, Spain (email: mfelipe@mat.upv.es)
*
✉For correspondence; e-mail: ealemany@mat.upv.es
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Abstract

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Let G be a finite p-solvable group. We prove that if G has exactly two conjugacy class sizes of p′-elements of prime power order, say 1 and m, then m=paqb, for two distinct primes p and q, and G either has an abelian p-complement or G=PQ×A, with P and Q a Sylow p-subgroup and a Sylow q-subgroup of G, respectively, and A is abelian. In particular, we provide a new extension of Itô’s theorem on groups having exactly two class sizes for elements of prime power order.

Keywords

finite groupsconjugacy class sizessolvable groups

MSC classification

Secondary: 20E45: Conjugacy classes
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 85 , Issue 3 , June 2012 , pp. 476 - 481
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2011

Footnotes

This research is supported by the Spanish Government, Proyecto MTM2010-19938-C03-02 and the second and third authors are supported by the Valencian Government, Proyecto PROMETEO/2011/30. The second author is also supported by grant Fundació Caixa-Castelló P11B2010-47.

References

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