Let G(q) be a finite Chevalley group, where q is a power of a good prime p, and let U(q) be a Sylow p-subgroup of G(q). Then a generalized version of a conjecture of Higman asserts that the number k(U(q)) of conjugacy classes in U(q) is given by a polynomial in q with integer coefficients. In an earlier paper, the first and the third authors developed an algorithm to calculate the values of k(U(q)). By implementing it into a computer program using GAP, they were able to calculate k(U(q)) for G of rank at most 5, thereby proving that for these cases k(U(q)) is given by a polynomial in q. In this paper we present some refinements and improvements of the algorithm that allow us to calculate the values of k(U(q)) for finite Chevalley groups of rank six and seven, except E_7. We observe that k(U(q)) is a polynomial, so that the generalized Higman conjecture holds for these groups. Moreover, if we write k(U(q)) as a polynomial in q-1, then the coefficients are non-negative.
Under the assumption that k(U(q)) is a polynomial in q-1, we also give an explicit formula for the coefficients of k(U(q)) of degrees zero, one and two.
It is proved that a profinite group $G$ has fewer than $2^{\aleph_0}$ conjugacy classes of $p$-elements for an odd prime $p$ if and only if its $p$-Sylow subgroups are finite. (Here, by a $p$-element one understands an element that either has $p$-power order or topologically generates a group isomorphic to ${\mathbb Z}_p$.) A weaker result is proved for $p=2$.
Let $F$ be a subfield of the algebraic closure of a finite field $\mathbb{F}_p$, $p \ne 2$, and let $R$ denote any ring such that $F[t] \subset R \subsetneq F(t)$. Let $G$ be a classical Chevalley group of adjoint type defined over $R$. We prove that the group $G(R)$ has the $R_{\infty}$-property.
Universidad Nacional de Córdoba（科尔多瓦国立大学）；CIEM-CONICET（数学与计算研究所-阿根廷国家科学研究委员会）；Vrije Universiteit Brussel（布鲁塞尔自由大学）；Southern University of Science and Technology（南方科技大学）；Università degli Studi di Padova（帕多瓦大学）
设$G$是一个有限群，cs $(G)$是$G$的共轭类大小的集合.回想$G$的元素$g$被称为\emph{vanishing element}，如果存在$G$的不可约特征标，在$g$上取值$0$，我们考虑cs $(G)$的一个特定子集，即集合vcs $(G)$，其元素是$G$的消失元素的共轭类大小。出于在\cite{BLP}的结果，我们描述了类的有限群$G$，使VCS $(G)$由一个单一的元素\emph{under the assumption that $G$ is supersolvable or $G$ has a normal Sylow $2$-subgroup}（特别是，奇数阶的群体）。作为一种特殊情况，我们还得到了共轭类大小为单零的有限群\emph{which is either a prime power or square-free}的一个刻划。
A group is called strongly bounded, if the speed with which it is generated by finitely many conjugacy classes has a positive, lower bound only dependent on the number of the conjugacy classes in question rather than the actual conjugacy classes. Earlier papers by Kedra, Libman and Martin and myself have shown that this is a property common to split Chevalley groups defined using an irreducible root system of rank at least $2$ and the ring of all S-algebraic integers and that the situation is dependent on the number theory of $R$ for ${\rm Sp}_4$ and $G_2.$ In this paper, we will show that ${\…
Let $G$ be a simple algebraic group over an algebraically closed field $k$ of characteristic $p$. The classification of the conjugacy classes of unipotent elements of $G(k)$ and nilpotent orbits of $G$ on $\operatorname{Lie}(G)$ is well-established. One knows there are representatives of every unipotent class as a product of root group elements and every nilpotent orbit as a sum of root elements. We give explicit representatives in terms of a Chevalley basis for the eminent classes. A unipotent (resp. nilpotent) element is said to be eminent if it is not contained in any subsystem subgroup (re…
The aim of this article is to explore global and local properties of finite groups whose integral group rings have only trivial central units, so-called cut groups. For such a group we study actions of Galois groups on its character table and show that the natural actions on the rows and columns are essentially the same, in particular the number of rational-valued irreducible characters coincides with the number of rational-valued conjugacy classes. Further, we prove a natural criterion for nilpotent groups of class 2 to be cut and give a complete list of simple cut groups. Also, the impact of…
