Related papers: $p$-Groups with Derived Length Three and Three Cha…

Answering Questions 19.23 and 19.24 from the Kourovka notebook we construct polycyclic groups with arbitrary torsion lengths and give examples of finitely presented groups whose quotients by their torsion subgroups are not finitely…

In the present paper, we shall show that for any prime number p, every finite p-group occurs as the Galois Group of the maximal unramified p-extension over a certain number field of finite degree. We shall also show that for any given…

Let k be an algebraically closed field of odd characteristic. We describe derivations of a large class of quantizations of affine normal Poisson varieties over k.

We present a basic pointwise bound for the irreducible characters of $\mathrm{SU}(3)$ and, as an application, derive new $L^p$ bounds for these characters. Our approach is based on the descent of characters to singular sets and the…

In the paper, a method of describing the outer derivations of the group algebra of a finitely presentable group is given. The description of derivations is given in terms of characters of the groupoid of the adjoint action of the group.

In this paper, we completely determine the irreducible characters of the four families of Suzuki $p$-groups.

For an irreducible complex character $\chi$ of a finite group $G$, the codegree of $\chi$ is defined by $|G:\ker(\chi)|/\chi(1)$, where $\ker(\chi)$ is the kernel of $\chi$. Given a prime $p$, we provide a classification of finite groups in…

We define and study supercharacters of the classical finite unipotent groups of symplectic and orthogonal types (over any finite field of odd characteristic). We show how supercharacters for groups of those types can be obtained by…

In this article, we use a unified approach to prove several classes of planar graphs are DP-$3$-colorable, which extend the corresponding results on $3$-choosability.

Let $D$ be an directed graph on $p\geq 10$ vertices with minimum degree at least $p-1$ and minimum semi-degree at least $ p/2 -1$. We present a detailed proof of the following result [13]: The digraph $D$ is pancyclic, unless some extremal…

We classify the finite groups whose non-linear irreducible characters that are not conjugate under the natural Galois action have distinct degrees, therefore extending the results in Berkovich et al. [Proc. Amer. Math. Soc. {\bf 115}…

Let $G$ be a finite group and let $\rm{Irr}(G)$ be the set of all irreducible complex characters of $G$. For a character $\chi \in \rm{Irr}(G)$, the number $\rm{cod}(\chi):=|G:\rm{ker}\chi|/\chi(1)$ is called the co-degree of $\chi$. The…

For a prime $p$, we determine a Sylow $p$-subgroup $D$ of a finite group $G$ such that the principal $p$-block $B$ of $G$ has four irreducible ordinary characters. It has been determined already for the cases where the number is up to three…

We classify pointed $p^3$-dimensional Hopf algebras $H$ over any algebraically closed field $k$ of prime characteristic $p>0$. In particular, we focus on the cases when the group $G(H)$ of group-like elements is of order $p$ or $p^2$, that…

We describe finite groups whose principal block contains only characters of prime power degree.

This thesis has three goals related to the automorphism groups of finite $p$-groups. The primary goal is to provide a complete proof of a theorem showing that, in some asymptotic sense, the automorphism group of almost every finite…

In this paper, we continue the classification work done in the first paper of the same name. With careful modifications of our previous approach, we are able to deduce (with two notable exceptions) which members of the previously introduced…

We ask which degree sequences admit a unique realisation as a $3$-polytopal graph (polyhedron) on $p$ vertices. We give an exhaustive list of these sequences for the case where one degree equals $p-1$ and exactly two or three of them equal…

A new conjecture on characters of finite groups, related to the McKay conjecture, was proposed recently by the first and third authors. In this paper, we prove it for $p$-solvable groups when $p$ is odd.

Recent examples of periodic bifurcations in descendant trees of finite p-groups with p in {2,3} are used to show that the possible p-class tower groups G of certain multiquadratic fields K with p-class group of type (2,2,2), resp. (3,3),…