we describe under various conditions abelian subgroups of the automorphism‎ ‎group $mathrm{aut}(t_{n})$ of the regular $n$-ary tree $t_{n}$‎, ‎which are‎ ‎normalized by the $n$-ary adding machine $tau =(e‎, ‎dots‎, ‎e,tau )sigma _{tau‎ ‎}$ where $sigma _{tau }$ is the $n$-cycle $left( 0,1‎, ‎dots‎, ‎n-1right) $‎. ‎as‎ ‎an application‎, ‎for $n=p$ a prime number‎, ‎and for $n=4$‎, ‎we prove that...

We fully characterize regular Hom-Lie structures on the incidence algebra I(X, K) of a finite connected poset X over field K. prove that such structure is sum central-valued linear map annihilating Jacobson radical with composition certain inner and multiplicative automorphisms K).

this article is intended to be a survey on some combinatorial topics in group theory. the bibliography at the end is neither claimed to be exhaustive, nor is it necessarily connected with a reference in the text. i include it as i see it revolves around the concepts which are discussed in the text.

Let G be a finite p-group of nilpotency class 2. We find necessary and sufficient conditions on G such that each central automorphism of G fixes the center of G element-wise.

Let T be an ergodic automorphism of the d-dimensional torus T, and f be a continuous function from T to R. On the probability space T equipped with the Lebesgue-Haar measure, we prove the weak convergence of the sequential empirical process of the sequence (f ◦T )i≥1 under some condition on the modulus of continuity of f . The proofs are based on new limit theorems and new inequalities for non-...

Let A be the flag complex of a generalised 2m-gon, with m 2 2, and let G be a group of special automorphisms of A. An automorphism of A is called a central root awomorphism if it fixes every vertex at distance less than or equal to m from a prescribed vertex of A. This paper begins a study of the structure of the pair (A, G), under the assumptions that G is finite and contains a central root au...

In this paper, we prove the existence of random Gibbs $ u $-state for partially hyperbolic system driven by an external force. Several examples such as fiber maps on tori, small perturbations systems, and composition automorphisms \mathbb{T}^3 with a fixed central direction are under consideration.