每个非交换的有限单群秩$n$在一个域的大小$q$，与可能的例外的Ree组$^2G_2(3^{2e+1})$，有一个表示与有限数量的发电机和关系和总长度$O(\log n +\log q)$。作为推论，我们推出了Holt的一个猜想：存在一个常数$C$，使得对每个有限单群$G$，每个素数$p$和每个不可约$F_p [G]$ -模$M$，有$\dim H^2(G,M)\leq C\dim M$.
英文摘要：
Every nonabelian finite simple group of rank $n$ over a field of size $q$, with the possible exception of the Ree groups $^2G_2(3^{2e+1})$, has a presentation with a bounded number of generators and relations and total length $O(\log n +\log q)$. As a corollary, we deduce a conjecture of Holt: there is a constant $C$ such that $\dim H^2(G,M)\leq C\dim M$ for every finite simple group $G$, every prime $p$ and every irreducible $F_p [G]$-module $M$.
We show that one can naturally describe elements of R. Thompson's finitely presented infinite simple group $V$, known by Thompson to have a presentation with four generators and fourteen relations, as products of permutations analogous to transpositions. This perspective provides an intuitive explanation towards the simplicity of $V$ and also perhaps indicates a reason as to why it was one of the first discovered infinite finitely presented simple groups: it is (in some basic sense) a relative of the finite alternating groups. We find a natural infinite presentation for $V$ as a group generate…
Guralnick, Kantor, Kassabov and Lubotzky (J. Eur. Math. Soc. 13.2, 2011, 391-458) [GKKL] give 3-generator 7-relator presentations of $A_n$ and $S_n$ with bit-length $O(\log n)$ for $n\geq5$. This is the best possible bit-length, since $Ω(\log n)$ bits are required to specify the integer $n$ in the input. However, the generators do not satisfy the relations. This paper considers the relevant arguments given in [GKKL], identifies where the errors occur, and shows how they can be fixed in order to recover this result. The presentations are available in Magma.
We exhibit a new presentation of the (equilateral) Von Dyck groups $D(2,3,n), \ n\ge 3$, in terms of two generators of order $n$ satisfying three relations, one of which is Artin's braid relation. By dropping the relation which fixes the order of the generators we obtain the universal covering groups of the corresponding Von Dyck groups. In the cases $n=3,\, 4,\,5$, these are respectively the double covers of the finite rotational tetrahedral, octahedral and icosahedral groups. When $n\ge 6$ we obtain infinite covers of the corresponding infinite Von Dyck groups. The interesting cases arise fo…
For each family of finite classical groups, and their associated simple quotients, we provide an explicit presentation on a specific generating set of size at most 8. Since there exist efficient algorithms to construct this generating set in any copy of the group, our presentations can be used to verify claimed isomorphisms between representations of the classical group. The presentations are available in Magma.
We give conditions on a presentation of a group, which imply that its Cayley complex is simplicial and the flag complex of the Cayley complex is systolic. We then apply this to Garside groups and Artin groups. We give a classification of the Garside groups whose presentation using the simple elements as generators satisfy our conditions. We then also give a dual presentation for Artin groups and identify in which cases the flag complex of the Cayley complex is systolic.
The higher-dimensional Thompson groups $nV$, for $n \geq 2$, were introduced by Brin in 2005. We provide new presentations for each of these infinite simple groups. The first is an infinite presentation, analogous to the Coxeter presentation for the finite symmetric group, with generating set equal to the set of transpositions in $nV$ and reflecting the self-similar structure of $n$-dimensional Cantor space. We then exploit this infinite presentation to produce further finite presentations that are considerably smaller than those previously known.
We give an infinite presentation for the mapping class group of a non-orientable surface. The generating set consists of all Dehn twists and all crosscap pushing maps along simple loops.
The problem of describing the group of units $\mathcal{U}(\mathbb{Z} G)$ of the integral group ring $\mathbb{Z} G$ of a finite group $G$ has attracted a lot of attention and providing presentations for such groups is a fundamental problem. Within the context of orders, a central problem is to describe a presentation of the unit group of an order $\mathcal{O}$ in the simple epimorphic images $A$ of the rational group algebra $\mathbb{Q} G$. Making use of the presentation part of Poincaré's Polyhedron Theorem, Pita, del Río and Ruiz proposed such a method for a large family of finite groups $G$ …
Tits has defined Kac-Moody and Steinberg groups over commutative rings, providing infinite dimensional analogues of the Chevalley-Demazure group schemes. Here we establish simple explicit presentations for all Steinberg and Kac-Moody groups whose Dynkin diagrams are hyperbolic and simply laced. Our presentations are analogues of the Curtis-Tits presentation of the finite groups of Lie type. When the ground ring is finitely generated, we derive the finite presentability of the Steinberg group, and similarly for the Kac-Moody group when the ground ring is a Dedekind domain of arithmetic type. Th…
We describe presentations of braid groups of type ADE and show how these presentations are compatible with mutation of quivers, building on work of Barot and Marsh for Coxeter groups. In types A and D these presentations can be understood geometrically using triangulated surfaces. We then give a categorical interpretation of the presentations, with the new generators acting as spherical twists at simple modules on derived categories of Ginzburg dg-algebras of quivers with potential.
This expository article revolves around the question to find short presentations of finite simple groups. This subject is one of the most active research areas of group theory in recent times. We bring together several known results on two-generation and $(2,3)$-generation of finite simple groups and how it impacts computational group theory.
For any root system and any commutative ring we give a relatively simple presentation of a group related to its Steinberg group St. This includes the case of infinite root systems used in Kac-Moody theory, for which the Steinberg group was defined by Tits and Morita-Rehmann. In most cases our group equals St, giving a presentation with many advantages over the usual presentation of St. This equality holds for all spherical root systems, all irreducible affine root systems of rank>2, and all 3-spherical root systems. When the coefficient ring satisfies a minor condition, the last condition can …
We augment the list of finite universal locally toroidal regular polytopes of type {3,3,4,3,3} due to P.McMullen and E.Schulte, adding as well as removing entries. This disproves a related long-standing conjecture. Our new universal polytope is related to a well-known Y-shaped presentation for the sporadic simple group $Fi_{22}$, and admits $S_4\times O_8^+(2){:}S_3$ as the automorphism group. We also discuss further extensions of its quotients in the context of Y-shaped presentations. As well, we note that two known examples of finite universal polytopes of type {3,3,4,3,3} are related to Y-s…
