Let G be a finite group and let d(G) be the minimal number of generators for G. It is well known that d(G) = 2 for all (non-abelian) finite simple groups. We prove that d(H) ≤ 4 for any maximal subgroup H of a finite simple group, and that this bound is best possible. We also investigate the random generation of maximal subgroups of simple and almost simple groups. By applying a recent theorem ...

We develop the local-global theory of blocks for profinite groups. Given a field k characteristic p and group G, one may express completed algebra k[[G]] as product ∏i∈IBi closed indecomposable algebras, called G. To each block B G we associate pro-p subgroup defect B, unique up to conjugacy in give several characterizations analogy with groups finite Our main theorem is version Brauer's first ...

In this paper, we present a classification of $$\mathbf{2}$$ -designs with $$\mathbf{gcd}({{\varvec{r}}},{\varvec{\lambda }})=\mathbf{1}$$ admitting flag-transitive automorphism groups. If $${{\varvec{G}}}$$ is group non-trivial -design $$\mathcal {{\varvec{D}}}$$ , then either $$(\mathcal {{\varvec{D}}},{{\varvec{G}}})$$ one the known examples described in or has $${{\varvec{q}}} = {{\varvec{p...

‎we characterize those groups $g$ and vector spaces $v$ such that $v$ is a faithful irreducible $g$-module and such that each $v$ in $v$ is centralized by a $g$-conjugate of a fixed non-identity element of the fitting subgroup $f(g)$ of $g$‎. ‎we also determine those $v$ and $g$ for which $v$ is a faithful quasi-primitive $g$-module and $f(g)$ has no regular orbit‎. ‎we do use these to show in ...

The vitamin D (1,25-dihydroxyvitamin D3) receptor (VDR) gene encodes a protein that functions in the transcriptional regulation of vitamin D-responsive genes and plays a role in innate immunity and adaptive immune responses. In this study, we investigated the relationship between VDR polymorphisms (BsmI, ApaI, and TaqI) and primary biliary cirrhosis (PBC) risk. We conducted an overall meta-anal...

Four avian metapneumovirus (AMPV) subgroups (A-D) have been reported previously based on genetic and antigenic differences. However, until now full length sequences of the only known isolates of European subgroup C and subgroup D viruses (duck and turkey origin, respectively) have been unavailable. These full length sequences were determined and compared with other full length AMPV and human me...

We investigate invariant random subgroups in groups acting on rooted trees. Let $\mathrm{Alt}_f(T)$ be the group of finitary even automorphisms $d$-ary tree $T$. prove that a nontrivial ergodic IRS acts without fixed points boundary $T$ contains level stabilizer, particular it is conjugate finite index subgroup. Applying technique to branch we an regular derived subgroup generalized rigid sta...

With any connected affine algebraic group G over an algebraically closed field K of characteristic zero, we associate another connected affine algebraic group D over K and a finite central subgroup F of D such that, up to isomorphism of algebraic groups, affine algebraic groups over K abstractly isomorphic to G are precisely of the form D/α(F )×Ks +, where α is an abstract automorphism of D and...

The distinguishing number D(G) of a graph G is the least integer d such that has vertex labeling with labels preserved only by trivial automorphism. Let ? be group acting on set X. for this action X, denoted D? (X), smallest natural k elements X can labeled so any label-preserving element fixes all x ? In particular, if faithful, then preserving identity. paper, we obtain an upper bound knowing...