Large characteristically simple sections of finite groups

In this paper we prove that if G is a group for which there are k non-Frattini chief factors isomorphic to a characteristically simple group A, then G has a normal section C/R that is the direct product of k minimal normal subgroups of G/R isomorphic to A. This is a significant extension of the notion of crown for isomorphic chief factors

Bounds on the number of maximal subgroups of finite groups: applications

The determination of bounds for the number of maximal subgroups of a given index in a finite group is relevant to estimate the number of random elements needed to generate a group with a given probability. In this paper, we obtain new bounds for the number of maximal subgroups of a given index in a finite group and we pin-point the universal constants that appear in some results in the literature related to the number of maximal subgroups of a finite group with a given index. This allows us to compare properly our bounds with some of the known bounds

Some Contributions to the Theory of Transformation Monoids

The aim of this paper is to present some contributions to the theory of finite transformation monoids. The dominating influence that permutation groups have on transformation monoids is used to describe and characterise transitive transformation monoids and primitive transitive transformation monoids. We develop a theory that not only includes the analogs of several important theorems of the classical theory of permutation groups but also contains substantial information about the algebraic structure of the transformation monoids. Open questions naturally arising from the substantial paper of Steinberg [A theory of transformation monoids: combinatorics and representation theory. Electron. J. Combin. 17 (2010), no. 1, Research Paper 164, 56 pp] have been answered. Our results can also be considered as a further development in the hunt for a solution of the Černý conjecture

Large maximal subgroups of finite almost simple groups

We prove in this paper that a finite almost simple group $R$ with socle the non-abelian simple group $S$ possesses a conjugacy class of maximal subgroups whose index coincides with the smallest index $\operatorname{l}(S)$ of a maximal group of $S$ or there exists a natural number $v_S \leq {\operatorname{l}(S)^2}$, depending only on $S$, such that $R$ has a conjugacy class of core-free maximal subgroups with index $v_S$. We show that the number of subgroups of the outer automorphism group of $S$ is bounded by $\log^3 {\operatorname{l}(S)}$ and $\operatorname{l}(S)^2 < |S|$.Comment: 17 page

Bounds on the number of maximal subgroups of finite groups

In this paper we obtain significant bounds for the number of maximal subgroups of a given index of a finite group. These results allow us to give new bounds for the number of random generators needed to generate a finite $d$-generated group with high probability.Comment: 18 page

SARS-CoV-2 infection in patients with melanoma: results of the Spanish Melanoma Group registry

Background The Spanish Melanoma Group (GEM) developed a national registry of patients with melanoma infected by SARS-CoV-2 ( GRAVID ).Methods The main objective was to describe the COVID-19 fatality rate in patients with melanoma throughout the pandemic, as well as to explore the effect of melanoma treatment and tumor stage on the risk of COVID-19 complications. These are the final data of the register, including cases from February 2020 to September 2021.Results One hundred-fifty cases were registered. Median age was 68 years (range 6-95), 61 (40%) patients were females, and 63 (42%) patients had stage IV. Thirty-nine (26%) were on treatment with immunotherapy, and 17 (11%) with BRAF-MEK inhibitors. COVID-19 was resolved in 119 cases, including 85 (57%) patients cured, 15 (10%) that died due to melanoma, and 20 (13%) that died due to COVID-19. Only age over 60 years, cardiovascular disorders, and diabetes mellitus increased the risk of death due to COVID-19, but not advanced melanoma stage nor melanoma systemic therapies. Three waves have been covered by the register: February-May 2020, August-November 2020, and December 2020-April 2021. The first wave had the highest number of registered cases and COVID-19 mortality.Conclusion Tumor stage or melanoma treatments are non-significant prognostic factors for COVID-19 mortality. During the pandemic in Spain there was a downward trend in the number of patients registered across the waves, as well as in the severity of the infection