相关论文: Presentations of Finite Simple Groups: Profinite a…

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…

How many generators and relations does ${\mathrm SL}_n({\mathbb F}_q[t, t^{-1}])$ need? In this paper we exhibit its explicit presentation with $9$ generators and $44$ relations. We investigate presentations of affine Kac-Moody groups over…

Let $G$ be the simple algebraic group $\mathrm{SL}_2$ defined over an algebraically closed field $k$ of characteristic $p > 0$. Using results of A. Parker, we develop a method which gives, for any $q \in \mathbb{N}$, a closed form…

In this paper we prove that the profinite completion $\mathcal{\hat G}$ of the Grigorchuk group $\mathcal{G}$ is not finitely presented as a profinite group. We obtain this result by showing that $H^2(\mathcal{\hat G},\field{F}_2)$ is…

For a group G we consider the set of natural numbers n for which the nth cohomology functor of G commutes with filtered colimit systems of coefficient modules. We find that for the large class of hierarchically decomposable groups there is…

We define, for any group $G$, finite approximations ; with this tool, we give a new presentation of the profinite completion $\hat{\pi} : G \to \hat{G}$ of an abtract group $G$. We then prove the following theorem : if $k$ is a finite prime…

All nonabelian finite simple groups of rank $n$ over a field of size $q$, with the possible exception of the Ree groups $^2G_2(3^{2e+1})$, have presentations with at most $80 $ relations and bit-length $O(\log n +\log q)$. Moreover, $A_n$…

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…

Let F_n be the free group on n generators, and P\Sigma_n be the group of automorphisms of F_n which send each generator to a conjugate of itself. Let K_n be the kernel of the homomorphism from P\Sigma_n to P\Sigma_{n-1} induced by mapping…

We construct the first example of a finitely-presented, residually-finite group that contains an infinite sequence of non-isomorphic finitely-presented subgroups such that each of the inclusion maps induces an isomorphism of profinite…

Assume $G$ is a solvable group whose elementary abelian sections are all finite. Suppose, further, that $p$ is a prime such that $G$ fails to contain any subgroups isomorphic to $C_{p^\infty}$. We show that if $G$ is nilpotent, then the…

Let $G$ be a finite group. In a famous article, Quillen describes an $\mathrm{F}$-isomorphism between commutative $\mathbb{N}$-graded $\mathbb{F}_{2}$-algebras $$\mathrm{q}_{G}:\mathrm{H}^{*}(G;\mathbb{F}_{2})\to\mathrm{L}(G)\ ,$$ with…

In this note we show that various (geometric/homological) finiteness properties are not profinite properties. For example for every $1 \le k, \ell \le \bbn$, there exist two finitely generated residually finite groups $\Ga_1$ and $\Ga_2$…

We prove that in every finitely generated profinite group, every subgroup of finite index is open; this implies that the topology on such groups is determined by the algebraic structure. This is deduced from the main result about finite…

Given any finitely presented group G we find a triangular algebra such that has two presentations, one with fundamental group G and another with trivial group. Thus proving that given a collection G1,...,Gn of finitely presented groups…

Let $G$ be a residually finite, good group of finite virtual cohomological dimension. We prove that the natural monomorphism $G\hookrightarrow\hat{G}$ induces a bijective correspondence between conjugacy classes of finite $p$-subgroups of…

We introduce various probablistic finiteness conditions for profinite groups related to positive finite generation (PFG). We investigate completed group rings which are PFG as modules, and use this to answer a question of Kionke and the…

We construct a correspondence between the cohomology groups of a group $G$ relative to a family of subgroups $\famS$ and the classes of `relative extensions' of $G$ by abelian groups, modulo a certain equivalence relation. We establish this…

We develop cohomological and homological theories for a profinite group $G$ with coefficients in the Pontryagin dual categories of pro-discrete and ind-profinite $G$-modules, respectively. The standard results of group (co)homology hold for…

We prove that the Drinfeld double of an arbitrary finite group scheme has finitely generated cohomology. That is to say, for G any finite group scheme, and D(G) the Drinfeld double of the group ring kG, we show that the self-extension…