Related papers: The z-Classes of Isometries

A computation shows that there are 77 (up to scalar shifts) possible pairs of integer coefficient polynomials of degree five, having roots of unity as their roots, and satisfying the conditions of Beukers and Heckman [1], so that the…

Noether, Fleischmann and Fogarty proved that if the characteristic of the underlying field does not divide the order $|G|$ of a finite group $G$, then the polynomial invariants of $G$ are generated by polynomials of degrees at most $|G|$.…

In this paper, we count the number of conjugacy and automorphism-conjugacy classes of elements of a ZM-group. The size of a conjugacy class with respect to these two equivalence relations is also counted.

Over an algebraically closed field we classify all minimal representation-infinite algebras where the lattice of two-sided ideals is not distributive. As a consequence there are only finitely many isomorphism classes of minimal…

Let $G$ be a connected, absolutely almost simple, algebraic group defined over a finitely generated, infinite field $K$, and let $\Gamma$ be a Zariski dense subgroup of $G(K)$. We show, apart from some few exceptions, that the…

Let $G$ be a finite group. An element $g$ of $G$ is called a vanishing element if there exists an irreducible character $\chi$ of $G$ such that $\chi(g) = 0$; in this case, we say that the conjugacy class of $g$ is a vanishing conjugacy…

It is known that that the centralizer of a matrix over a finite field depends, up to conjugacy, only on the type of the matrix, in the sense defined by J. A. Green. In this paper an analogue of the type invariant is defined that in general…

In this paper we classify the finite groups satisfying the following property $P_4$: their orders of representatives are set-wise relatively prime for any 4 distinct non-central conjugacy classes.

We give a geometric characterization of finite rational groups. In particular, we prove that a finite group is rational if and only if there exists a finite geometry $\Gamma$ of type $I$ and action of $G$ on $\Gamma$ as a group of…

Let F be characteristic zero field, G a residually finite group and W a G-prime and PI F-algebra. By constructing G-graded central polynomials for W, we prove the G-graded version of Posner's theorem. More precisely, if S denotes all…

Let k be a finite field, a global field or a local non-archimedean field. Let H_1 and H_2 be two split, connected, semisimple algebraic groups defined over k. We prove that if H_1 and H_2 share the same set of maximal k-tori up to…

In this paper, we give the enumeration of z-classes in finite Coxeter groups.

A finite group of order divisible by 3 in which centralizers of 3-elements are 3-subgroups will be called a C{\theta}{\theta}-group. The prime graph (or Gruenberg-Kegel graph) of a finite group G is denoted by {\Gamma}(G) (or GK(G)) and its…

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…

We show that if a complex has free finitely generated reduced homology groups for two consecutive dimensions and trivial homology for all other dimensions, then it must have the homotopy type of a wedge of spheres of two consecutive…

We give canonical matrices of a pair (A,B) consisting of a nondegenerate form B and a linear operator A satisfying B(Ax,Ay)=B(x,y) on a vector space over F in the following cases: (i) F is an algebraically closed field of characteristic…

The palindromic automorphism group is a subgroup of the automorphism group $Aut(F_3).$ We establish a necessary and sufficient condition for a matrix in $GL_n(\mathbb{Z})$ representing a palindromic automorphism of $F_n.$ We prove that the…

Let $\V$ be a vector space over a field $\F$. Assume that the characteristic of $\F$ is \emph{large}, i.e. $char(\F)>\dim \V$. Let $T: \V \to \V$ be an invertible linear map. We answer the following question in this paper: When does $\V$…

In \cite{1}, a class of gyrogroups $(G,\odot)$ has been associated to certain groups $(G,\cdot)$. We give a necessary and sufficient condition for $(G,\odot)$ to be gyrocommutative. We also prove that under a suitable assumption two finite…

The finite-degree Zariski (Z-) closure is a classical algebraic object, that has found a key place in several applications of the polynomial method in combinatorics. In this work, we characterize the finite-degree Z-closures of a subclass…