论文标题
在最大数量的元素成对生成有限交流组的元素数量上
On the maximal number of elements pairwise generating the finite alternating group
论文作者
论文摘要
令$ g $为$ n $的交替组。令$ω（g）$为$ g $的子集$ s $的最大大小，以便$ \ langle x，y \ rangle = g $每当$ x，y in s $和$ x \ neq y $和$ x \ neq y $和$σ（g）$时，$ c $的最低符号是$ g $的$ g $。我们证明，当$ n $因复合数字而变化时，$σ（g）/ω（g）$倾向于$ 1 $ as $ n \ to \ to \ infty $。此外，我们明确计算$σ（a_n）$，$ n \ geq 21 $一致至$ 3 $ modulo $ 18 $。
Let $G$ be the alternating group of degree $n$. Let $ω(G)$ be the maximal size of a subset $S$ of $G$ such that $\langle x,y \rangle = G$ whenever $x,y \in S$ and $x \neq y$ and let $σ(G)$ be the minimal size of a family of proper subgroups of $G$ whose union is $G$. We prove that, when $n$ varies in the family of composite numbers, $σ(G)/ω(G)$ tends to $1$ as $n \to \infty$. Moreover, we explicitly calculate $σ(A_n)$ for $n \geq 21$ congruent to $3$ modulo $18$.
