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Any sufficiently often differentiable curve in the orbit space $V/G$ of a real finite-dimensional orthogonal representation $G \to O(V)$ of a finite group $G$ admits a differentiable lift into the representation space $V$ with locally…

We obtain a Lie theoretic intrinsic characterization of the connected and simply connected solvable Lie groups whose regular representation is a factor representation. When this is the case, the corresponding von Neumann algebras are…

Let $F$ be a finite field of odd cardinality $q$, $A=F[T]$ the polynomial ring over $F$, $k=F(T)$ the rational function field over $F$ and $\mathcal{H}$ the set of square-free monic polynomials in $A$ of degree odd. If $D\in\mathcal{H}$, we…

We obtain a complete classification of the continuous unitary representations of oligomorphic permutation groups (those include the infinite permutation group $S_\infty$, the automorphism group of the countable dense linear order, the…

We show that there exists a family of groups $G_n$ and nontrivial irreducible representations $\rho_n$ such that, for any constant $t$, the average of $\rho_n$ over $t$ uniformly random elements $g_1, \ldots, g_t \in G_n$ has operator norm…

We introduce the notion of stable representations, -- it is a new class of the representations of a certain class of groups which defined with positive definite functions which generalize the classical notion of the characters (or trace).…

We study a conjecture by Deaconescu on the solubility of finite groups with claims that if more than half of the elements in a finite group has the same order $k$, then the group is soluble. We show that the original conjecture fails by…

Let G be a 2-group of order 2^n, n>5, and nilpotency class n-2. The invariants of such groups determined by their group algebras over the field of two elements are given in the paper.

We describe complete simplicial toric varieties on which a unipotent group acts with a finite number of orbits. We also provide a complete list of such varieties in the case where the dimension is equal to 2.

In this article we introduce order preserving representations of fundamental groups of surfaces into Lie groups with bi-invariant orders. By relating order preserving representations to weakly maximal representations, introduced in…

A finite presentation < X | R > of a finite group is called `just finite' if removing any relation from R results in a presentation for an infinite group. It has been an open question (Kourovka Notebook, Problem 21.10) whether every finite…

We classify finite primitive permutation groups having a suborbit of length 5. As a corollary, we obtain a classification of finite vertex-primitive graphs of valency 5. In the process, we also classify finite almost simple groups that have…

We show that if $G$ is a finite group whose Sylow $2$-subgroups are wreathed, then the intersection $\Outc(G) \cap \OutCol(G)$ has odd order, where $\Outc(G)$ and $\OutCol(G)$ denote the class-preserving and Coleman outer automorphism…

We give new classes of examples of orbits of the diagonal group in the space of unit volume lattices in R^d for d > 2 with nice (homogeneous) orbit closures, as well as examples of orbits with explicitly computable but irregular orbit…

In this article we introduce and study a class of finite groups for which the orders of normal subgroups satisfy a certain inequality. It is closely connected to some well-known arithmetic classes of natural numbers.

Let $G$ be a finite group and let $p$ be a prime. In this paper, we study the structure of finite groups with a large number of $p$-regular conjugacy classes or, equivalently, a large number of irreducible $p$-modular representations. We…

We show that the minimum number of vertices of a simplicial complex with fundamental group $\mathbb{Z}^{n}$ is at most $O(n)$ and at least $\Omega(n^{3/4})$. For the upper bound, we use a result on orthogonal 1-factorizations of $K_{2n}$.…

In this paper we prove that a finite group of order $r$ has at most $$ 7.3722\cdot r^{\frac{\log_2r}{4}+1.5315}$$ subgroups.

We construct irreducible unitary representations of a finitely generated free group which are weakly contained in the left regular representation and in which a given linear combination of the generators has an eigenvalue. When the…

We show that the local equivalence problem for second-order ordinary differential equations under point transformations is completely characterized by differential invariants of order at most 10 and that this upper bound is sharp. We also…