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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

The number of Hall $\pi$-subgroups of a $\pi$-separable group


Author: Alexandre Turull
Journal: Proc. Amer. Math. Soc. 132 (2004), 2563-2565
MSC (2000): Primary 20D20
DOI: https://doi.org/10.1090/S0002-9939-04-07412-X
Published electronically: March 3, 2004
MathSciNet review: 2054781
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Abstract: We observe a simple formula to compute the number $\nu_\pi(G)$ of Hall $\pi$-subgroups of a $\pi$-separable finite group $G$ in terms of only the action of a fixed Hall $\pi$-subgroup of $G$ on a set of normal $\pi'$-sections of $G$. As a consequence, we obtain that $\nu_\pi(K)$divides $\nu_\pi(G)$ whenever $K$ is a subgroup of a finite $\pi$-separable group $G$. This generalizes a recent result of Navarro. In addition, our method gives an alternative proof of Navarro's result.


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Additional Information

Alexandre Turull
Affiliation: Department of Mathematics, University of Florida, Gainesville, Florida 32611-8105
Email: turull@math.ufl.edu

DOI: https://doi.org/10.1090/S0002-9939-04-07412-X
Received by editor(s): February 17, 2003
Received by editor(s) in revised form: June 7, 2003
Published electronically: March 3, 2004
Additional Notes: The author was partially supported by an NSA Grant
Communicated by: Jonathan I. Hall
Article copyright: © Copyright 2004 American Mathematical Society