Related papers: Arbitrary p-Gradient Values

We study dp-minimal infinite profinite groups that are equipped with a uniformly definable fundamental system of open subgroups. We show that these groups have an open subgroup $A$ such that either $A$ is a direct product of countably many…

Let $A$ be an Artin group. A partition $\mathcal{P}$ of the set of standard generators of $A$ is called admissible if, for all $X,Y \in \mathcal{P}$, $X \neq Y$, there is at most one pair $(s,t) \in X \times Y$ which has a relation. An…

We obtain an asymptotic upper bound for the product of the $p$-parts of the orders of certain composition factors of a finite group acting completely reducibly and faithfully on a finite vector space of order divisible by a prime $p$. An…

We introduce a special class of powerful $p$-groups that we call powerfully nilpotent groups that are finite $p$-groups that possess a central series of a special kind. To these we can attach the notion of a powerful nilpotence class that…

We construct an infinite finitely generated recursively presented residually finite algorithmically finite group $G$ answering thereby a question of Myasnikov and Osin. Moreover, $G$ is "very infinite" and "very algorithmically finite" in…

For a commutative finite $\mathbb{Z}$-algebra, i.e., for a commutative ring $R$ whose additive group is finitely generated, it is known that the group of units of $R$ is finitely generated, as well. Our main results are algorithms to…

Let p be a prime number. We give the explicit structure of 2- nilpotent multiplier for each finite 2-generator p-group of class two. Moreover, 2-capable groups in that class are characterized.

Let $q$ be a power of a prime $p$, $G$ be a finite abelian group, where $p$ does not divide $|G|$,and let $n$ be a positive integer. In this paper we find a formula for the number of irreducible representations of $G$ of a given dimension…

We prove two approximations of the open problem of whether the adjoint group of a non-nilpotent nil ring can be finitely generated: We show that the adjoint group of a non-nilpotent Jacobson radical cannot be boundedly generated, and on the…

A finitely generated solvable group with unbounded iterated identity is constructed.

We prove that for every prime $p$ algebraically clean graphs of groups are virtually residually $p$-finite and cohomologically $p$-complete. We also prove that they are cohomologically good. We apply this to certain $2$-dimensional Artin…

Let O be the ring of S-integers in a number field k. We prove that if the group of units O^* is infinite then every matrix in $\Gamma$ = SL_2(O) is a product of at most 9 elementary matrices. This completes a long line of research in this…

The girth of a finitely generated group G is the supremum of the girth of Cayley graphs for G over all finite generating sets. Let G be a finitely generated subgroup of the mapping class group Mod(S), where S is a compact orientable…

We study the existence of (unmixed) Beauville structures in finite $p$-groups, where $p$ is a prime. First of all, we extend Catanese's characterisation of abelian Beauville groups to finite $p$-groups satisfying certain conditions which…

We construct, for the first time, various types of specific non-special finite $p$-groups having abelian automorphism group. More specifically, we construct groups $G$ with abelian automorphism group such that $\gamma_2(G) < \mathrm{Z}(G) <…

The mod-p cohomology ring of a non-trivial finite p-group is an infinite dimensional, finitely presented graded unital algebra over the field with p elements, with generators in positive degrees. We describe an effective algorithm to test…

We prove that for every finitely generated group $\Gamma$, at least one of the following holds: (1) $\Gamma$ has fixed price; (2) each of its Cayley graphs $G$ has infinitely many infinite clusters for some Bernoulli percolation on $G$.

Every finite group $G$ has a normal series each of whose factors either is soluble or is a direct product of nonabelian simple groups. We define the nonsoluble length $\lambda (G)$ as the minimum number of nonsoluble factors in a series of…

We construct uncountably categorical 3-nilpotent groups of exponent p > 3. They are not one-based and do not allow the interpretation of an infinite field. Therefore they are counterexamples to Zilbers Conjecture. First 2-nilpotent new…

Consider $\operatorname{Sym}(n)$, endowed with the normalized Hamming metric $d_n$. A finitely-generated group $\Gamma$ is \emph{P-stable} if every almost homomorphism $\rho_{n_k}\colon \Gamma\rightarrow\operatorname{Sym}(n_k)$ (i.e., for…