arXiv subjects

Saveliy Skresanov

Publications and source records attributed to Saveliy Skresanov.

2 recordsLinked to original sources

arXiv2019

Group topologies on integers and S-unit equations

A sequence of integers $ \{ s_n \}_{n \in \mathbb{N}} $ is called a T-sequence if there exists a Hausdorff group topology on $ \mathbb{Z} $ such that $ \{ s_n \}_{n \in \mathbb{N}} $ converges to zero. For every finite set of primes $ S $ we build a Hausdorff group topology on $ \mathbb{Z} $ such that every growing sequence of $ S $-integers converges to zero. As a corollary, we solve in the affirmative an open problem by I.V. Protasov and E.G. Zelenuk asking if $ \{ 2^n + 3^n \}_{n \in \mathbb{N}} $ is a T-sequence. Our results rely on a nontrivial number-theoretic fact about $ S $-unit equations.

math.GR↗

arXiv2019

The proper definition and Wielandt-Hartley's theorem for submaximal $\mathfrak{X}$-subgroups

A nonempty class $\mathfrak{X}$ of finite groups is called complete if it is closed under taking subgroups, homomorphic images and extensions. We deal with a classical problem of determining $\mathfrak{X}$-maximal subgroups. We consider two definitions of submaximal $\mathfrak{X}$-subgroups suggested by Wielandt and discuss which one better suits our task. We prove that these definitions are not equivalent yet Wielandt-Hartley's theorem holds true for either definition of $\mathfrak{X}$-submaximality. We also give some applications of the strong version of Wielandt-Hartley's theorem.

math.GR↗