论文标题
关于有限组的常见横向概率
On the common transversal probability in finite groups
论文作者
论文摘要
令$ g $为有限的组，让$ h $为$ g $的子组。我们计算以$ p_g（h）$表示的概率，$ g $中的$ h $的左横向也是正确的横向，因此是双面的。此外，我们定义并用$ \ mathrm {tp}（g）$表示，$ g $的常见横向概率是最低限度的，取代了$ g $ $ g $的所有子组，$ g $，$ p_g（h）$。我们证明了有关不变$ \ mathrm {tp}（g）$的许多结果，例如下限和上限，并且可能达到的值。我们还表明，$ \ mathrm {tp}（g）$确定$ g $的结构属性。最后，提出和讨论了几个开放问题。
Let $G$ be a finite group, and let $H$ be a subgroup of $G$. We compute the probability, denoted by $P_G(H)$, that a left transversal of $H$ in $G$ is also a right transversal, thus a two-sided one. Moreover, we define, and denote by $\mathrm{tp}(G)$, the common transversal probability of $G$ to be the minimum, taken over all subgroups $H$ of $G$, of $P_G(H)$. We prove a number of results regarding the invariant $\mathrm{tp}(G)$, like lower and upper bounds, and possible values it can attain. We also show that $\mathrm{tp}(G)$ determines structural properties of $G$. Finally, several open problems are formulated and discussed.
