Primitive permutation groups as products of point stabilizers

We prove that there exists a universal constant c such that any finite primitive permutation group of degree n with a non-trivial point stabilizer is a product of no more than clog⁡n point stabilizers

Genetic Characterization of Cancer of Unknown Primary Using Liquid Biopsy Approaches

Cancers of unknown primary (CUPs) comprise a heterogeneous group of rare metastatic tumors whose primary site cannot be identified after extensive clinical\u2013pathological investigations. CUP patients are generally treated with empirical chemotherapy and have dismal prognosis. As recently reported, CUP genome presents potentially druggable alterations for which targeted therapies could be proposed. The paucity of tumor tissue, as well as the difficult DNA testing and the lack of dedicated panels for target gene sequencing are further relevant limitations. Here, we propose that circulating tumor cells (CTCs) and circulating tumor DNA (ctDNA) could be used to identify actionable mutations in CUP patients. Blood was longitudinally collected from two CUP patients. CTCs were isolated with CELLSEARCH\uae and DEPArrayTM NxT and Parsortix systems, immunophenotypically characterized and used for single-cell genomic characterization with Ampli1TM kits. Circulating cell-free DNA (ccfDNA), purified from plasma at different time points, was tested for tumor mutations with a CUP-dedicated, 92-gene custom panel using SureSelect Target Enrichment technology. In parallel, FFPE tumor tissue was analyzed with three different assays: FoundationOne CDx assay, DEPArray LibPrep and OncoSeek Panel, and the SureSelect custom panel. These approaches identified the same mutations, when the gene was covered by the panel, with the exception of an insertion in APC gene. which was detected by OncoSeek and SureSelect panels but not FoundationOne. FGFR2 and CCNE1 gene amplifications were detected in single CTCs, tumor tissue, and ccfDNAs in one patient. A somatic variant in ARID1A gene (p.R1276 17) was detected in the tumor tissue and ccfDNAs. The alterations were validated by Droplet Digital PCR in all ccfDNA samples collected during tumor evolution. CTCs from a second patient presented a pattern of recurrent amplifications in ASPM and SEPT9 genes and loss of FANCC. The 92-gene custom panel identified 16 non-synonymous somatic alterations in ccfDNA, including a deletion (I1485Rfs 1719) and a somatic mutation (p. A1487V) in ARID1A gene and a point mutation in FGFR2 gene (p.G384R). Our results support the feasibility of non-invasive liquid biopsy testing in CUP cases, either using ctDNA or CTCs, to identify CUP genetic alterations with broad NGS panels covering the most frequently mutated genes

Histone modifications underlie monocyte dysregulation in patients with systemic sclerosis, underlining the treatment potential of epigenetic targeting.

Background and objective S ystemic sclerosis (SSc) is a severe autoimmune disease, in which the pathogenesis is dependent on both genetic and epigenetic factors. Altered gene expression in SSc monocytes, particularly of interferon (IFN)-responsive genes, suggests their involvement in SSc development. We investigated the correlation between epigenetic histone marks and gene expression in SSc monocytes. Methods C hromatin immunoprecipitation followed by sequencing (ChIPseq) for histone marks H3K4me3 and H3K27ac was performed on monocytes of nine healthy controls and 14 patients with SSc. RNA sequencing was performed in parallel to identify aberrantly expressed genes and their correlation with the levels of H3K4me3 and H3K27ac located nearby their transcription start sites. ChIP-qPCR assays were used to verify the role of bromodomain proteins, H3K27ac and STATs on IFNresponsive gene expression. Results 1046 and 534 genomic loci showed aberrant H3K4me3 and H3K27ac marks, respectively, in SSc monocytes. The expression of 381 genes was directly and significantly proportional to the levels of such chromatin marks present near their transcription start site. Genes correlated to altered histone marks were enriched for immune, IFN and antiviral pathways and presented with recurrent binding sites for IRF and STAT transcription factors at their promoters. IFN\u3b1 induced the binding of STAT1 and STAT2 at the promoter of two of these genes, while blocking acetylation readers using the bromodomain BET family inhibitor JQ1 suppressed their expression. Conclusion SS c monocytes have altered chromatin marks correlating with their IFN signature. Enzymes modulating these reversible marks may provide interesting therapeutic targets to restore monocyte homeostasis to treat or even prevent SSc

Direct products of finite groups as unions of proper subgroups

We determine all the ways in which a direct product of two finite groups can be expressed as the set-theoretical union of proper subgroups in a family of minimal cardinality

Finite groups, minimal bases and the intersection number

Let G be a finite group and recall that the Frattini subgroup Frat(G) is the intersection of all the maximal subgroups of G. In this paper, we investigate the intersection number of G, denoted alpha(G), which is the minimal number of maximal subgroups whose intersection coincides with Frat(G). In earlier work, we studied alpha(G) in the special case where G is simple and here we extend the analysis to almost simple groups. In particular, we prove that alpha(G) <= 4 for every almost simple group G, which is best possible. We also establish new results on the intersection number of arbitrary finite groups, obtaining upper bounds that are defined in terms of the chief factors of the group. Finally, for almost simple groups G we present best possible bounds on a related invariant beta(G), which we call the base number of G. In this setting, beta(G) is the minimal base size of G as we range over all faithful primitive actions of the group and we prove that the bound beta(G) <= 4 is optimal. Along the way, we study bases for the primitive action of the symmetric group S-ab on the set of partitions of [1, ab] into a parts of size a >= b, determining the exact base size for a b. This extends earlier work of Benbenishty, Cohen and Niemeyer

The first families of highly symmetric Kirkman Triple Systems whose orders fill a congruence class

Kirkman triple systems (KTSs) are among the most popular combinatorial designs and their existence has been settled a long time ago. Yet, in comparison with Steiner triple systems, little is known about their automorphism groups. In particular, there is no known congruence class representing the orders of a KTS with a number of automorphisms at least close to the number of points. We partially fill this gap by proving that whenever v≡ 39 (mod 72), or v≡ 4 e48 + 3 (mod 4 e96) and e≥ 0 , there exists a KTS on v points having at least v- 3 automorphisms. This is only one of the consequences of an investigation on the KTSs with an automorphism group G acting sharply transitively on all but three points. Our methods are all constructive and yield KTSs which in many cases inherit some of the automorphisms of G, thus increasing the total number of symmetries. To obtain these results it was necessary to introduce new types of difference families (the doubly disjoint ones) and difference matrices (the splittable ones) which we believe are interesting by themselves

The first families of highly symmetric Kirkman Triple Systems whose orders fill a congruence class

Kirkman triple systems (KTSs) are among the most popular combinatorial designs and their existence has been settled a long time ago. Yet, in comparison with Steiner triple systems, little is known about their automorphism groups. In particular, there is no known congruence class representing the orders of a KTS with a number of automorphisms at least close to the number of points. We partially fill this gap by proving that whenever v≡ 39 (mod 72), or v≡ 4 e48 + 3 (mod 4 e96) and e≥ 0 , there exists a KTS on v points having at least v- 3 automorphisms. This is only one of the consequences of an investigation on the KTSs with an automorphism group G acting sharply transitively on all but three points. Our methods are all constructive and yield KTSs which in many cases inherit some of the automorphisms of G, thus increasing the total number of symmetries. To obtain these results it was necessary to introduce new types of difference families (the doubly disjoint ones) and difference matrices (the splittable ones) which we believe are interesting by themselves