论文标题
简单群体和贝特曼的命令 - 顽强的猜想
Orders of simple groups and the Bateman--Horn Conjecture
论文作者
论文摘要
我们使用贝特曼（Bateman） - 数字理论中的猜想来给出有力的证据，证明彼得·诺伊曼（Peter Neumann）的问题有积极的答案，是否有无限的六个素数产物的无限简单组。 （在1890年代，伯恩赛德，弗罗贝尼乌斯和霍尔德分类的人都被归类。）满足这种情况的组为$ {\ rm psl} _2（8）$，$ {\ rm psl} _2（\ rm psl} _2（9）$和$ {\ rm psl} $ _2（p）$ primes $ primes $ p.该猜想表明，通过提供分布的启发式估算，与计算机搜索的证据紧密吻合，这是无限的许多这样的素数。我们还简要讨论了该猜想在群体理论中的其他问题的应用，例如置换组的分类和线性质量的线性组的分类，有限简单组的功率图的结构以及高度对称的块设计的构建。
We use the Bateman--Horn Conjecture from number theory to give strong evidence of a positive answer to Peter Neumann's question, whether there are infinitely many simple groups of order a product of six primes. (Those with fewer than six were classified by Burnside, Frobenius and Hölder in the 1890s.) The groups satisfying this condition are ${\rm PSL}_2(8)$, ${\rm PSL}_2(9)$ and ${\rm PSL}_2(p)$ for primes $p$ such that $p^2-1$ has just six prime factors. The conjecture suggests that there are infinitely many such primes, by providing heuristic estimates for their distribution which agree closely with evidence from computer searches. We also briefly discuss the applications of this conjecture to other problems in group theory, such as the classifications of permutation groups and of linear groups of prime degree, the structure of the power graph of a finite simple group, and the construction of highly symmetric block designs.
