Central Schur rings over the projective special linear groups

In [1 Bannai, E. (1991). Subschemes of some association schemes. J. Algebra 14:167–188.[Crossref], [Web of Science ®] , [Google Scholar]], Bannai presents a fusion condition and uses this to consider central Schur rings (S-rings) over the simple groups PSL(2,q) where q is a prime power. In this paper, we concretely describe all such S-rings in terms of symmetric S-rings over cyclic groups. The final section discusses counting these.     

Symmetric designs and projective special linear groups of dimension at most four

In this article, we study symmetric $\left(v,k,\lambda \right)$ designs admitting a flag‐transitive and point‐primitive automorphism group $G$ whose socle is isomorphic to a projective special linear group of dimension at most four. We, in particular, determine all such possible parameters $\left(v,k,\lambda \right)$ and show that such a design belongs to one of two infinite families of point‐hyperplane designs or it is isomorphic to a design with parameters (7, 3, 1), (7, 4, 2), (11, 5, 2), (11, 6, 3), (15, 8, 4), (35, 18, 9), or (56, 45, 18).     

Recognition of the Projective Special Linear Group over GF（3）

Let P be a finite group and denote by w（P） the set of its element orders. P is called k-recognizable by the set of its element orders if for any finte group G with ω（G） =ω（P） there are, up to isomorphism, k finite groups G such that G ≌P. In this paper we will prove that the group Lp（3）, where p 〉 3 is a prime number, is at most 2-recognizable.     

Periodic subgroups of projective linear groups in positive characteristic

We classify the maximal irreducible periodic subgroups of PGL(q, ), where is a field of positive characteristic p transcendental over its prime subfield, q = p is prime, and ^× has an element of order q. That is, we construct a list of irreducible subgroups G of GL(q, ) containing the centre ^×1 _q of GL(q, ), such that G/ ^×1 _q is a maximal periodic subgroup of PGL(q, ), and if H is another group of this kind then H is GL(q, )-conjugate to a group in the list. We give criteria for determining when two listed groups are conjugate, and show that a maximal irreducible periodic subgroup of PGL(q, ) is self-normalising.      

Pure quantitative characterization of finite projective special unitary groups

We prove that each projective special unitary group G can be characterized using only the set of element orders of G and the order of G.     

Generalizations of Fibonacci polynomials and free linear groups

The aim of this paper is to relate some generalized Fibonacci polynomials to the problem of knowing when a group (or semigroup) generated by some linear transformations is free.     

A free product of finitely generated nilpotent groups amalgamating a cycle that is not subgroup separable

We exhibit a counterexample to a recent assertion concerning the subgroup separability of groups in the title. The example also serves as a simplification of work of Gitik and Rips.

     

Sets of elements that pairwise generate a linear group

Let G be any of the groups (P)GL(n,q), (P)SL(n,q). Define a (simple) graph Γ=Γ(G) on the set of elements of G by connecting two vertices by an edge if and only if they generate G. Suppose that n is at least 12. Then the maximum size of a complete subgraph in Γ is equal to the chromatic number of Γ if , or if , q is odd and G=(P)SL(n,q). This work was motivated by a question of Blackburn.     

Rationality of quotients by linear actions of affine groups

LetG = SLn(C)Cn be the (special) affine group. In this paper we study the representation theory of G and in particular the question of rationality for V/G, where V is a generically free G-representation. We show that the answer to this question is positive (Theorem 6.1) if the dimension of V is sufficiently large and V is indecomposable. We explicitly characterize two-step extensions 0 → S → V → Q → 0, with completely reducible S and Q, whose rationality cannot be obtained by the methods presented here (The...     

从特殊线性群到一般射影线性群的同态的一个性质

设F,K为域,GLn（F）,SLn（F）分别表示F上的n级一般线生群和n级特殊线性群．PGLn（F）,PSLn（F）分别表示F上的n级射影一般线性群和n级射影特殊线性群．φ：SLn（F）→PGLn（K）,n≥3为非平凡同态．本文确定了当K的持征为2时η的—个性质．     

交换环上特殊线性李代数的极大子代数

文章利用有单位元且2,3是单位的交换环的极大理想刻画了其上特殊线性李代数包含典范环面的极大子代数.确定了特殊线性李代数极大子代数的个数,并证明了每个极大子代数均可通过置换矩阵共轭于标准的极大子代数.     

Sets of Transvections Generating Subgroups Isomorphic to Special Linear Groups

ABSTRACT

The main result of this paper is a graph-theoretic necessary and sufficient condition, for a given set of transvections in SL(n, K) (n > 2 and K a finite field of characteristic not 2 or 3), to generate a group isomorphic to SL (m, L), for some m and some subfield L of K.     

On the torsion units of the integral group ring of finite projective special linear groups

H. J. Zassenhaus conjectured that any unit of finite-order and augmentation one in the integral group ring of a finite group G is conjugate in the rational group algebra to an element of G. One way to verify this is showing that such unit has the same distribution of partial augmentations as an element of G and the HeLP Method provides a tool to do that in some cases. In this paper, we use the HeLP Method to describe the partial augmentations of a hypothetical counterexample to the conjecture for the projective special linear groups.     

Subgroup separability of certain generalized free products of nilpotent-by-finite groups

In this paper, we show that certain generalized free products of nilpotent-by-finite groups are subgroup separable when the amalgamated subgroup is h× D where D is in the center of both factors.     

Groups of p-absolute Galois type that are not absolute Galois groups

Let p be a prime. We study pro-p groups of p-absolute Galois type, as defined by Lam–Liu–Sharifi–Wake–Wang. We prove that the pro-p completion of the right-angled Artin group associated to a chordal simplicial graph is of p-absolute Galois type, and moreover it satisfies a strong version of the Massey vanishing property. Also, we prove that Demushkin groups are of p-absolute Galois type, and that the free pro-p product — and, under certain conditions, the direct product — of two pro-p groups of p-absolute Galois type satisfying the Massey vanishing property, is again a pro-p group of p-absolute Galois type satisfying the Massey vanishing property. Consequently, there is a plethora of pro-p groups of p-absolute Galois type satisfying the Massey vanishing property that do not occur as absolute Galois groups.     

Generic representations of the finite general linear groups and the Steenrod algebra: II

The category of generic representations over the finite fieldF _q , used in PartI to study modules over the Steenrod algebra, is here related to the modular representation theory of the groups GL _n (F _q ). This leads to a simple and elegant approach to the classic objects of study: irreducible representations, extensions of modules, homology stability, etc. Connections to current research in algebraicK-theory involving stableK-theory and Topological Hochschild Homology are also explained.Partially funded by the NSF.     

Order of elements in the groups related to the general linear group

For a natural number n and a prime power q the general, special, projective general and projective special linear groups are denoted by GL_n(q), SL_n(q), PGL_n(q) and PSL_n(q), respectively. Using conjugacy classes of elements in GL_n(q) in terms of irreducible polynomials over the finite field GF(q) we demonstrate how the set of order elements in GL_n(q) can be obtained. This will help to find the order of elements in the groups SL_n(q), PGL_n(q) and PSL_n(q). We also show an upper bound for the order of elements in SL_n(q).