Base sizes of primitive permutation groups

Let G be a permutation group, acting on a set $$\varOmega $$ Ω of size n. A subset $${\mathcal {B}}$$ B of $$\varOmega $$ Ω is a base for G if the pointwise stabilizer $$G_{({\mathcal {B}})}$$ G ( B ) is trivial. Let b(G) be the minimal size of a base for G. A subgroup G of $$\mathrm {Sym}(n)$$ Sym ( n ) is large base if there exist integers m and $$r \ge 1$$ r ≥ 1 such that $${{\,\mathrm{Alt}\,}}(m)^r \unlhd G \le {{\,\mathrm{Sym}\,}}(m)\wr {{\,\mathrm{Sym}\,}}(r)$$ Alt ( m ) r ⊴ G ≤ Sym ( m ) ≀ Sym ( r ) , where the action of $${{\,\mathrm{Sym}\,}}(m)$$ Sym ( m ) is on k-element subsets of $$\{1,\dots ,m\}$$ { 1 , ⋯ , m } and the wreath product acts with product action. In this paper we prove that if G is primitive and not large base, then either G is the Mathieu group $$\mathrm {M}_{24}$$ M 24 in its natural action on 24 points, or $$b(G)\le \lceil \log n\rceil +1$$ b ( G ) ≤ ⌈ log n ⌉ + 1 . Furthermore, we show that there are infinitely many primitive groups G that are not large base for which $$b(G) > \log n + 1$$ b ( G ) > log n + 1 , so our bound is optimal.

NEW CONSTRUCTIONS OF SELF-COMPLEMENTARY CAYLEY GRAPHS

Vertex-primitive self-complementary graphs were proved to be affine or in product action by Guralnick et al. [‘On orbital partitions and exceptionality of primitive permutation groups’, Trans. Amer. Math. Soc.356 (2004), 4857–4872]. The product action type is known in some sense. In this paper, we provide a generic construction for the affine case and several families of new self-complementary Cayley graphs are constructed.

Ranks, Subdegrees and Suborbital graphs of the product action of Affine Groups

The action of affine groups on Galois field has been studied. For instance, studied the action of on Galois field for a power of prime. In this paper, the rank and subdegree of the direct product of affine groups over Galois field acting on the cartesian product of Galois field is determined. The application of the definition of the product action is used to achieve this. The ranks and subdegrees are used in determination of suborbital graph, the non-trivial suborbital graphs that correspond to this action have been constructed using Sims procedure and were found to have a girth of 0, 3, 4 and 6.

Probiotic complex Anzymesporin when rearing and fattening of young pigs

Feed additives and means of probiotic and prebiotic action, enzyme drugs, therapeutic and prophylactic additives of complex action aimed at stimulating non-specific immunity, prevention and treatment of mixed gastrointestinal infections and digestive disorders are an alternative to antibiotics. Domestic scientists have developed a new feed probiotic, Enzymsporin, which contains a complex of lyophilized spore-forming bacteria Bacillus subtilis and Bacillus licheniformis in a concentration of 5×10 9 CFU/g, which causes a wide range of product action against pathogenic and conditionally pathogenic microorganisms. The purpose of the work was to study a new probiotic complex with antibacterial and immunomodulatory properties as an alternative to the use of antibiotic-containing drugs in feeding of growing and fattening young pigs. The effectiveness and feasibility of using the probiotic complex Enzymsporin in the diets of growing and fattening young pigs in a comparative aspect with the antibiotic-containing drug Virginiamycin has been studied. It has been proved on the base of zootechnical, physiological, biochemical and economic research methods that the optimal dosage for the introduction of the probiotic complex Enzymsporin into compound feed for growing young pigs is 0,5 kg/t. We recommend pig breeding farms to input in the diet of growing and fattening young pigs probiotic complex Enzymsporin in the amount of 0,5 kg/t, which helps to stimulate metabolism and leads to better use of feed, increases the livability of livestock and economically justified. The use of probiotic complex has a pronounced biological effect in the growth and livability of piglets during rearing, contributes to increasing the profitability of pig production.

TENSOR-PRODUCT COACTION FUNCTORS

Recent work by Baum et al. [‘Expanders, exact crossed products, and the Baum–Connes conjecture’, Ann. K-Theory 1(2) (2016), 155–208], further developed by Buss et al. [‘Exotic crossed products and the Baum–Connes conjecture’, J. reine angew. Math. 740 (2018), 111–159], introduced a crossed-product functor that involves tensoring an action with a fixed action $(C,\unicode[STIX]{x1D6FE})$ , then forming the image inside the crossed product of the maximal-tensor-product action. For discrete groups, we give an analogue for coaction functors. We prove that composing our tensor-product coaction functor with the full crossed product of an action reproduces their tensor-crossed-product functor. We prove that every such tensor-product coaction functor is exact, and if $(C,\unicode[STIX]{x1D6FE})$ is the action by translation on $\ell ^{\infty }(G)$ , we prove that the associated tensor-product coaction functor is minimal, thereby recovering the analogous result by the above authors. Finally, we discuss the connection with the $E$ -ization functor we defined earlier, where $E$ is a large ideal of $B(G)$ .

Hereditarily just infinite profinite groups with complete Hausdorff dimension spectrum

We prove that the inverse limits of certain iterated wreath products in product action have complete Hausdorff dimension spectrum with respect to their unique maximal filtration of open normal subgroups. Moreover we can produce explicitly subgroups with a specified Hausdorff dimension.