Related papers: On products of conjugacy classes in general linear…

In this paper, we study a class of generalized intersection matrix Lie algebras $\gim(M_n)$, and prove that its every finite-dimensional semi-simple quotient is of type $M(n,{\bf a}, {\bf c},{\bf d})$. Particularly, any finite dimensional…

Let $M$ be a factor with separable predual and $G$ a compact group of automorphisms of $M$ whose action is minimal, i.e. $M^{G^\prime}\cap M = C$, where $M^G$ denotes the $G$-fixed point subalgebra. Then every intemediate von Neumann…

Given a conjugacy class $\mathcal{C}$ in a group $G$ we define a new graph, $\Gamma(\mathcal{C})$, whose vertices are elements of $\mathcal{C}$; two vertices $g,h\in \mathcal{C}$ are connected in $\Gamma(\mathcal{C})$ if $[g,h]=1$ and…

Let $N \subset M$ be an irreducible inclusion of type type II$_1$ factors with finite Jones index. We shall introduce the notion of normality for intermediate subfactors of the inclusion $N \subset M$. If the depth of $N \subset M$ is 2,…

For a graph $H$, let $c(H)=\inf\{c\,:\,e(G)\geq c|G| \mbox{ implies } G\succ H\,\}$, where $G\succ H$ means that $H$ is a minor of $G$. We show that if $H$ has average degree $d$, then $$ c(H)\le (0.319\ldots+o_d(1))|H|\sqrt{\log d} $$…

We prove that there is an absolute constant $c>0$ with the following property: if $Z/pZ$ denotes the group of prime order $p$, and a subset $A\subset Z/pZ$ satisfies $1<|A|<p/2$, then for any positive integer…

Maximal connected grading classes of a finite-dimensional algebra $A$ are in one-to-one correspondence with Galois covering classes of $A$ which admit no proper Galois covering and therefore are key in computing the intrinsic fundamental…

We classify $\mathcal{R}$- and $\mathcal{L}$-cross-sections of wreath products of finite inverse symmetric semigroups $\mathcal{IS}_m \wr_p \mathcal{IS}_n$ up to isomorphism. We show that every isomorphism of $\mathcal{R}$ ($\mathcal{L}$-)…

We study finite groups $G$ with elements $g$ such that $\lvert \mathbf{C}_G(g)\rvert = \lvert G:G' \rvert$. (Such elements generalize fixed-point-free automorphisms of finite groups.) We show that these groups have a unique conjugacy class…

Every finite group whose order is divisible by a prime $p$ has at least $2 \sqrt{p-1}$ conjugacy classes.

Let $q$ be an odd prime, $m$ a positive integer, and let $\Ga_m(q)$ be the group generated by two elements $x$ and $y$ subject to the relations $x^{2m}=y^{qm}=1$ and $x^2=y^q$; that is, $\Ga_m(q)$ is the free product of two cyclic groups of…

Let $m\in\mathbb{Z}$ be an integer and $L_m=\mathbb{Q}(\alpha)$ be the simplest cubic field with class number $h_m$ and conductor $\mathfrak{f}_m$ where $\alpha$ is a root of $f_m(X)=X^3-mX^2-(m+3)X-1$. Let $\mathcal{O}_{L_m}$ be the ring…

In this note, we construct a family of semisimple Hopf algebras $H_{n,m}$ of dimension $n^m m!$ over a field of characteristic zero containing a primitive $n$th root of unity, where $n, m \geq 2$ are integers. The well-known…

Given a positive integer $N$ and real number $\alpha\in [0, 1]$, let $m(\alpha,N)$ denote the minimum, over all sets $A\subset \mathbb{Z}/N\mathbb{Z}$ of size at least $\alpha N$, of the normalized count of 3-term arithmetic progressions…

In this paper, we classify conjugacy classes of centralizers of irreducible subgroups in $PSL(n,\mathbb{C})$ using alternate modules a.k.a. finite abelian groups with an alternate bilinear form. When $n$ is squarefree, we prove that these…

Let G be an almost simple reductive group with Weyl group W. Let B be a Borel subgroup of G. Let C be an elliptic conjugacy class in W and let w be an element of minimal length of C. We investigate the existence of a semisimple class of G…

Let $G$ be a finite $p$-group, where $p$ is a prime number, and $a\in G$. Denote by $\Cl(a)=\{gag^{-1}\mid g\in G\}$ the conjugacy class of $a$ in $G$. Assume that $|\Cl(a)|=p^n$. Then $\Cl(a)\Cl(a^{-1})=\{xy\mid x\in \Cl(a), y\in…

An element of a group is called $\textit{strongly reversible}$ or $\textit{strongly real}$ if it can be expressed as a product of two involutions. We provide necessary and sufficient conditions for an element of $\mathrm{SL}(n,\mathbb{C})$…

We prove a central limit theorem (CLT) for the product of a class of random singular matrices related to a random Hill's equation studied by Adams$\unicode{x2013}$Bloch$\unicode{x2013}$Lagarias. The CLT features an explicit formula for the…

Assume $\mathcal{C}$ is the class of all linear orders $L$ such that $L$ is not a countable union of well ordered sets, and every uncountable subset of $L$ contains a copy of $\omega_1$. We show it is consistent that $\mathcal{C}$ has…