论文标题
在最大数量的元素数量上，成对生成对称群的偶数
On the maximal number of elements pairwise generating the symmetric group of even degree
论文作者
论文摘要
令$ g $为对称的学位$ n $。令$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 $。我们证明，这两个函数$σ（g）$和$ω（g）$在$ \ frac {1} {2} {2} \ binom {n} {n/2} $时，当$ n $偶数时。这与S. Blackburn的结果一起，意味着$σ（g）/ω（g）$倾向于$ 1 $ as $ n \ to \ infty $。此外，我们在$ω（g）$上给出$（1-O（1））n $的下限，该$与有限简单组的分类无关。我们还计算出足够大的$ n $的图表的集合数，定义为：顶点是$ g $的元素，如果$ \ langle x，y \ rangle \ geq a_n $，则将两个顶点$ x，y $连接。
Let $G$ be the symmetric 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 both functions $σ(G)$ and $ω(G)$ are asymptotically equal to $\frac{1}{2} \binom{n}{n/2}$ when $n$ is even. This, together with a result of S. Blackburn, implies that $σ(G)/ω(G)$ tends to $1$ as $n \to \infty$. Moreover, we give a lower bound of $(1-o(1))n$ on $ω(G)$ which is independent of the classification of finite simple groups. We also calculate, for large enough $n$, the clique number of the graph defined as follows: the vertices are the elements of $G$ and two vertices $x,y$ are connected by an edge if $\langle x,y \rangle \geq A_n$.
