Abstract
In this note, I propose the following conjecture: a finite group \(G\) is nilpotent if and only if its largest quotient \(B\)-group \(\beta (G)\) is nilpotent. I give a proof of this conjecture under the additional assumption that \(G\) be solvable. I also show that this conjecture is equivalent to the following: the kernel of restrictions to nilpotent subgroups is a biset-subfunctor of the Burnside functor.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Baumann, M.: The composition factors of the functor of permutation modules. J. Algebra 344, 284–295 (2011)
Bouc, S.: Foncteurs d’ensembles munis d’une double action. J. Algebra 183(0238), 664–736 (1996)
Bouc, S.: Biset functors for finite groups, volume 1990 of Lecture Notes in Mathematics. Springer, Berlin (2010)
Gluck, D.: Idempotent formula for the Burnside ring with applications to the \(p\)-subgroup simplicial complex. Ill. J. Math. 25, 63–67 (1981)
Gorenstein, W.: Finite groups. Chelsea, London (1968)
Yoshida, T.: Idempotents in Burnside rings and Dress induction theorem. J. Algebra 80, 90–105 (1983)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Bouc, S. A conjecture on \(B\)-groups. Math. Z. 274, 367–372 (2013). https://doi.org/10.1007/s00209-012-1074-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00209-012-1074-0