Let (At | i e I) be an indexed family of nonempty intervals of a linearly ordered set {L, <). Let (/, E) be the intersection graph of (̂ 4, | I G / ) , that is {i,j}eE if and only if At n At J= 0 . In [6], R. Rado considered the following sentence R(K), where K is a cardinal number. If Chr(J, E n |V]) ^ K for all ; £ / , |J| ^ K + , then Chr(/ , E) < K. He proved (see [6, Theorem 2]) that R(K) h...

Let $\pi$ be a proper subset of the set all primes. Denote by $r$ smallest prime which does not belong to and $m = r$ if $r 2$ or $3$ r-1$ \geqslant 5$. We study following conjecture: conjugacy class $D$ finite group $G$ is contained in $\pi$-radical $\mathrm{O}_\pi(G)$ only every $m$ elements generate $\pi$-subgroup. confirm this conjecture for each whose nonabelian composition factors are iso...

We discuss an open problem on the discreteness of subgroups of (SL2(R)) (n ≥ 2) generated by n linearly independent upper triangular matrices and n linearly independent lower triangular matrices. According to a conjecture by Margulis, only Hilbert modular groups can arise this way. The purpose of this note is to explain how this open problem is related to another conjecture on the orbit behavio...

We classify the filled groups of order pqr for primes p, q and r. Our result confirms a conjecture Anabanti, Erskine Hart these examples soluble groups.

It is shown that all nontrivial elements in higher K-groups of toric varieties over a class of regular rings are annihilated by iterations of the natural Frobenius type endomorphisms. This is a higher analog of the triviality of vector bundles on affine toric varieties. 1. Statement of the main result The nilpotence conjecture in K-theory of toric varieties, treated in our previous works, asser...

vasil&apos;ev posed problem 16.26 in [the kourovka notebook: unsolved problems in group theory, 16th ed.,sobolev inst. math., novosibirsk (2006).] as follows:does there exist a positive integer $k$ such that there are no $k$ pairwise nonisomorphicnonabelian finite simple groups with the same graphs of primes? conjecture: $k = 5$.in [zvezdina, on nonabelian simple groups having the same prime gr...