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Register a new userDeciding if a variety forms an algebraic group
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Let $n$ be a positive integer and let $f_1, ldots, f_r$ be polynomials in $n^2$ indeterminates over an algebraically closed field $K$. We describe an algorithm to decide if the invertible matrices contained in the variety of $f_1, ldots, f_r$ form a subgroup of $GL(n,K)$; that is, we show how to decide if the polynomials $f_1, ldots, f_r$ define a linear algebraic group.
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A Cayley graph for a group $G$ is CCA if every automorphism of the graph that preserves the edge-orbits under the regular representation of $G$ is an element of the normaliser of $G$. A group $G$ is then said to be CCA if every connected Cayley graph on $G$ is CCA. We show that a finite simple group is CCA if and only if it has no element of order 4. We also show that many 2-groups are non-CCA.
Let k be a global field and let k_v be the completion of k with respect to v, a non-archimedean place of k. Let mathbf{G} be a connected, simply-connected algebraic group over k, which is absolutely almost simple of k_v-rank 1. Let G=mathbf{G}(k_v). Let Gamma be an arithmetic lattice in G and let C=C(Gamma) be its congruence kernel. Lubotzky has shown that C is infinite, confirming an earlier conjecture of Serre. Here we provide complete solution of the congruence subgroup problem for Gamm$ by determining the structure of C. It is shown that C is a free profinite product, one of whose factors is hat{F}_{omega}, the free profinite group on countably many generators. The most surprising conclusion from our results is that the structure of C depends only on the characteristic of k. The structure of C is already known for a number of special cases. Perhaps the most important of these is the (non-uniform) example Gamma=SL_2(mathcal{O}(S)), where mathcal{O}(S) is the ring of S-integers in k, with S={v}, which plays a central role in the theory of Drinfeld modules. The proof makes use of a decomposition theorem of Lubotzky, arising from the action of Gamma on the Bruhat-Tits tree associated with G.
In this paper, we attempt to develop the Quillen Suslin theory for the algebraic fundamental group of a ring. We give a surjective group homomorphism from the algebraic fundamental group of the field of the real numbers to the group of integers. At the end of the paper, we also propose some problems related to the algebraic fundamental group of some particular type of rings.
The paper surveys several results on the topology of the space of arcs of an algebraic variety and the Nash problem on the arc structure of singularities.
Let $G$ be a simple algebraic group over an algebraically closed field $k$, where $mathrm{char}, k$ is either 0 or a good prime for $G$. We consider the modality $mathrm{mod}(B : mathfrak u)$ of the action of a Borel subgroup $B$ of $G$ on the Lie algebra $mathfrak u$ of the unipotent radical of $B$, and report on computer calculations used to show that $mathrm{mod}(B:mathfrak u) = 20$, when $G$ is of type $mathrm E_8$. This completes the determination of the values for $mathrm{mod}(B:mathfrak u)$ for $G$ of exceptional type.
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