Group extensions and graphs

NOTICE: this is the author’s version of a work that was accepted for publication in Expositiones Mathematicae. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Expositiones Mathematicae, [Volume 34, Issue 3, 2016, Pages 327-334] DOI#10.1016/j.exmath.2015.07.005¨A classical result of Gaschütz affirms that given a finite A-generated group G and a prime p, there exists a group G# and an epimorphism phi: G# ---> G whose kernel is an elementary abelian p-group which is universal among all groups satisfying this property. This Gaschütz universal extension has also been described in the mathematical literature with the help of the Cayley graph. We give an elementary and self-contained proof of the fact that this description corresponds to the Gaschütz universal extension. Our proof depends on another elementary proof of the Nielsen-Schreier theorem, which states that a subgroup of a free group is free.This work has been supported by the grant MTM-2014-54707-C3-1-P of the Ministerio de Economia y Competitividad (Spain). The first author is also supported by Project No. 11271085 from the National Natural Science Foundation of China. The second author is supported by the predoctoral grant AP2010-2764 (Programa FPU, Ministerio de Educacion, Spain).Ballester Bolinches, A.; Cosme-Llópez, E.; Esteban Romero, R. (2016). Group extensions and graphs. Expositiones Mathematicae. 34(3):327-334. https://doi.org/10.1016/j.exmath.2015.07.005S32733434

On finite T-groups

This paper has been published in Journal of the Australian Mathematical Society 75(2):181-192 (2003). The final publication is available at Cambridge University Press Journals, http://journals.cambridge.org/abstract_S1446788700003712 http://dx.doi.org/10.1017/S1446788700003712[EN] Characterisations of finite groups in which normality is a transitive relation are presented in the paper. We also characterise the finite groups in which every subgroup is either permutable or coincides with its permutiser as the groups in which every subgroup is permutable.Supported by Proyecto PB97-0674-C02-02 and Proyecto PB97-O6O4 from DGICYT, Ministerio de Education y Cienciahttp://journals.cambridge.org/abstract_S1446788700003712Ballester Bolinches, A.; Esteban Romero, R. (2003). On finite T-groups. Journal of the Australian Mathematical Society. 2(75). doi:10.1017/S144678870000371227

On self-normalising subgroups of finite groups

The aim of this paper is to characterise the classes of groups in which every subnormal subgroup is normal, permutable, or S-permutable by the embedding of the subgroups (respectively, subgroups of prime power order) in their normal, permutable, or S-permutable closure, respectively

Counting homomorphisms onto finite solvable groups

We present a method for computing the number of epimorphisms from a finitely-presented group G to a finite solvable group \Gamma, which generalizes a formula of G\"aschutz. Key to this approach are the degree 1 and 2 cohomology groups of G, with certain twisted coefficients. As an application, we count low-index subgroups of G. We also investigate the finite solvable quotients of the Baumslag-Solitar groups, the Baumslag parafree groups, and the Artin braid groups.Comment: 30 pages; accepted for publication in the Journal of Algebr

On lower bounds for the Ihara constants A(2) and A(3)

Let X be a curve over the finite field of q elements and let N(X), g(X) be its number of rational points and genus respectively. The Ihara constant A(q) is defined by the limit superior of N(X)/g(X) as the genus of X goes to infinity. In this paper, we employ a variant of Serre's class field tower method to obtain an improvement of the best known lower bounds on A(2) and A(3).Comment: 22 pages; to appear in Compos. Mat

Compression of Finite Group Actions and Covariant Dimension, II

Let $G$ be a finite group and $\phi : V\to W$ an equivariant morphism of finite dimensional $G$-modules. We say that $\phi$ is faithful if $G$ acts faithfully on $\phi(V)$. The covariant dimension of $G$ is the minimum of the dimension of $\bar{\phi(V)}$ taken over all faithful $\phi$. In \cite{KS07} we investigated covariant dimension and were able to determine it in many cases. Our techniques largely depended upon finding homogeneous faithful covariants. After publication of \cite{KS07}, the junior author of this article pointed out several gaps in our proofs. Fortunately, this inspired us to find better techniques, involving multihomogeneous covariants, which have enabled us to extend and complete the results, simplify the proofs and fill the gaps of \cite{KS07}

On a class of p-soluble groups

Electronic version of an article published as Algebra Colloquium, 12(2)(2005), 263-267 DOI: 10.1142/S1005386705000258. © copyright World Scientific Publishing Company. http://www.worldscientific.com/doi/abs/10.1142/S1005386705000258[EN] Let p be a prime. The class of all p-soluble groups G such that every p-chief factor of G is cyclic and all p-chief factors of G are G-isomorphic is studied in this paper. Some results on T-, PT-, and PST -groups are also obtained.Supported by Grant BFM2001-1667-C03-03, MCyT (Spain) and FEDER (European Union)http://www.worldscientific.com/doi/abs/10.1142/S1005386705000258Ballester Bolinches, A.; Esteban Romero, R.; Pedraza Aguilera, MC. (2005). On a class of p-soluble groups. Algebra Colloquium. 2(12). doi:10.1142/S100538670500025821

Sylow permutable subnormal subgroups of finite groups

Paper published by Elsevier in J. Algebra, 215(2):727-738 (2002). The final publication is available at www.sciencedirect.com. http://dx.doi.org/10.1006/jabr.2001.9138[EN] An extension of the well-known Frobenius criterion of p-nilpotence in groups with modular Sylow p-subgroups is proved in the paper. This result is useful to get information about the classes of groups in which every subnormal subgroup is permutable and Sylow permutable.Supported by Proyecto PB97-0674 and Proyecto PB97-0604-C02-02 from DGICYT, Ministerio de Educaci´on y CienciaBallester Bolinches, A.; Esteban Romero, R. (2002). Sylow permutable subnormal subgroups of finite groups. Journal of Algebra. 2(251). doi:10.1006/jabr.2001.9138225