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The number of subgroups and the number of cyclic subgroups are natural combinatorial invariants of a finite group. We investigate how restrictions on these quantities, together with the number of distinct prime divisors of $|G|$, enforce…

Using percolation techniques, Gaboriau and Lyons recently proved that every countable, discrete, nonamenable group $\Gamma$ contains measurably the free group $\mathbf F_2$ on two generators: there exists a probability measure-preserving,…

Let $H$ be a subgroup of a group $G$. We say that $H$ satisfies the power condition with respect to $G$, or $H$ is a power subgroup of $G$, if there exists a non-negative integer $m$ such that $H=G^{m}=<g^{m} | g \in G >$. In this note, the…

Let $\mathcal M$ be a weakly o-minimal non-valuational structure, and $\mathcal N$ its canonical o-minimal extension (by Wencel). We prove that every group $G$ definable in $\mathcal M$ is a subgroup of a group $K$ definable in $\mathcal…

An action of a group $G$ is highly transitive if $G$ acts transitively on $k$-tuples of distinct points for all $k \geq 1$. Many examples of groups with a rich geometric or dynamical action admit highly transitive actions. We prove that if…

Tannaka's Theorem states that a linear algebraic group G is determined by the category of finite dimensional G-modules and the forgetful functor. We extend this result to linear differential algebraic groups by introducing a category…

For a finite group $G$, we define the inclusion graph of subgroups of $G$, denoted by $\mathcal I(G)$, is a graph having all the proper subgroups of $G$ as its vertices and two distinct vertices $H$ and $K$ in $\mathcal I(G)$ are adjacent…

We give sufficient conditions for a linear differential equation to have a given semisimple group as its Galois group. For any linear algebraic group G given as a semidirect product of a finite subgroup and a normal subgroup that is a…

A 2-group is a "categorified" version of a group, in which the underlying set G has been replaced by a category and the multiplication map has been replaced by a functor. Various versions of this notion have already been explored; our goal…

In this paper, we prove certain multiplicity one theorems and define twisted gamma factors for irreducible generic cuspidal representations of split $G_2$ over finite fields $k$ of odd characteristic. Then we prove the first converse…

Let $G$ be a group. Write $G^{*}=G\setminus \{1\}$. An element $x$ of $G^{*}$ will be called deficient if $ \langle x\rangle < C_G(x)$ and it will be called non-deficient if $\langle x\rangle = C_G(x).$ If $x\in G$ is deficient…

We define $G$-cospectrality of two $G$-gain graphs $(\Gamma,\psi)$ and $(\Gamma',\psi')$, proving that it is a switching isomorphism invariant. When $G$ is a finite group, we prove that $G$-cospectrality is equivalent to cospectrality with…

This paper is concerned with extensions of geometric stability theory to some nonelementary classes. We prove the following theorem: Theorem: Let C be a large homogeneous model of a stable diagram D. Let p, q in S_D(A), where p is…

The codegree of an irreducible character $\chi$ of a finite group $G$ is defined as $|G:\ker\chi|/\chi(1)$. The codegree graph $\Gamma(G)$ of a finite group $G$ is the graph whose vertices are the prime divisors of $|G|$, where two distinct…

Let G be the free product of groups A and B with commuting subgroups H \leqslant A and K \leqslant B, and let C be the class of all finite groups or the class of all finite p-groups. We derive the description of all C-separable cyclic…

Let $\Gamma$ be a centerless irreducible higher rank arithmetic lattice in characteristic zero. We prove that if $\Gamma$ is either non-uniform or is uniform of orthogonal type and dimension at least 9, then $\Gamma$ is bi-interpretable…

Following the literature, a group $G$ is called a group of central type if $G$ has an irreducible character that vanishes on $G\setminus Z(G)$. Motivated by this definition, we say that a character $\chi\in {\rm Irr}(G)$ has central type if…

We first formulate a general scheme for the classification of 2-compact groups in terms of maximal torus normalizer pairs. Applying this scheme, we show that all connected and some non-connected 2-compact groups are N-determined. We also…

Let $W$ be a Coxeter group whose proper parabolic subgroups are finite. According to Theorem~1.12 of [1], if the module of a finite $W$-digraph $\Gamma$ is isomorphic to the module of a $W$-graph over $Q$, then $\Gamma$ is acyclic. We…

The Farrell-Jones and the Baum-Connes Conjecture say that one can compute the algebraic K- and L-theory of the group ring and the topological K-theory of the reduced group C^*-algebra of a group G in terms of these functors for the…