Related papers: Upper bounds on the number of conjugacy classes in…

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…

Landau's theorem on conjugacy classes asserts that there are only finitely many finite groups, up to isomorphism, with exactly $k$ conjugacy classes for any positive integer $k$. We show that, for any positive integers $n$ and $s$, there…

We classify all real and strongly real classes of the finite special unitary group $SU_n(q)$. Unless $q \equiv 3 (mod 4)$ and $n |4$, the classification of real classes is similar to that of the finite special linear group $SL_n(q)$. We…

Let $U=U_n(q)$ be the group of lower unitriangular $n \times n$-matrices with entries in the field $\mathbb F_q$ with $q$ elements for some prime power $q$ and $n \in \mathbb N$. We investigate the restriction to $U$ of the permutation…

Let $\|A\|_{p,q}$ be the norm induced on the matrix $A$ with $n$ rows and $m$ columns by the H\"older $\ell_p$ and $\ell_q$ norms on $R^n$ and $R^m$ (or $C^n$ and $C^m$), respectively. It is easy to find an upper bound for the ratio…

We consider the groups of regular circulant matrices over finite fields and integer residue class rings. In both cases we present a formula for the order of these groups. We also make a first step towards finding the algebraic structure of…

We give a quadratic lower bound on the dimension of the space of conjugacy classes of subgroups of SL(n,R) that are limits under conjugacy of the diagonal subgroup. We give the first explicit examples of abelian n-1 dimensional subgroups of…

We obtain new uniform upper bounds for the (non necessarily symmetric) tensor rank of the multiplication in the extensions of the finite fields $\F_q$ for any prime or prime power $q\geq2$; moreover these uniform bounds lead to new…

We give an upper bound on the set of spherical classes in $H_*QX$ when $X = P,S^1$. This is related to the Curtis conjecture on spherical classes in $H_*Q_0S^0$. The results also provide some control over the bordism classes on of…

We classify the parabolic unitals in regular nearfield planes of odd order $q^2$ whose linear collineation group has the maximal size of $q^3-q$. We also establish a number of more general results concerning parabolic unitals in regular…

We study the conjugacy classes of the classical affine groups. We derive generating functions for the number of classes analogous to formulas of Wall and the authors for the classical groups. We use these to get good upper bounds for the…

Combinatorics on multisets is used to deduce new upper and lower bounds on the number of numerical semigroups of each given genus, significantly improving existing ones. In particular, it is proved that the number $n_g$ of numerical…

Let $G$ be a finite group and $\pi$ be a set of primes. We show that if the number of conjugacy classes of $\pi$-elements in $G$ is larger than $5/8$ times the $\pi$-part of $|G|$ then $G$ possesses an abelian Hall $\pi$-subgroup which…

We determine the minimal number of separating invariants for the invariant ring of a matrix group $G < \mathrm{GL}_n(\mathbb{F}_q)$ over the finite field $\mathbb{F}_q$. We show that this minimal number can be obtained with invariants of…

We exhibit an explicit formula for the cardinality of solutions to a class of quadratic matrix equations over finite fields. We prove that the orbits of these solutions under the natural conjugation action of the general linear groups can…

We extend to an arbitrary number field the best known bounds towards the Ramanujan conjecture for the groups GL(n), n=2, 3, 4. In particular, we present a technique which overcomes the analytic obstacles posed by the presence of an infinite…

For a free group $\mathbb{F}$ of finite rank such that $\text{rank}(\mathbb{F})\geq 3$, we prove that the set of weak limits of a conjugacy class in $\mathbb{F}$ under iterates of some hyperbolic $\phi\in\mathsf{Out(\mathbb{F})}$ is equal…

We characterize the group property of being with infinite conjugacy classes (or icc, in which all conjugacy classes beside 1 are infinite) for extensions of some specific groups ; namely extensions of abelian, centerless, icc, or word…

Let $p\equiv -q \equiv 5\pmod 8$ be two prime integers. In this paper, we investigate the unit groups of the fields $ L_1 =\mathbb{Q}(\sqrt 2, \sqrt{p}, \sqrt{q}, \sqrt{-1} )$ and $ L_1^+=\mathbb{Q}(\sqrt 2, \sqrt{p}, \sqrt{q} )$.…

Let $G$ be a finite group. Let $k(G)$ denote the number of conjugacy classes of $G$ and let $m(G)$ denote the least positive integer $n$ such that the union of any $n$ distinct non-trivial conjugacy classes of $G$ together with the identity…