We consider the problem of discovering subgroup $H$ of permutation group $S_{n}$. Unlike the traditional $H$-invariant networks wherein $H$ is assumed to be known, we present a method to discover the underlying subgroup, given that it satisfies certain conditions. Our results show that one could discover any subgroup of type $S_{k} (k \leq n)$ by learning an $S_{n}$-invariant function and a linear transformation. We also prove similar results for cyclic and dihedral subgroups. Finally, we provide a general theorem that can be extended to discover other subgroups of $S_{n}$. We also demonstrate the applicability of our results through numerical experiments on image-digit sum and symmetric polynomial regression tasks.

翻译：我们考虑发现置换群 $S_{n}$ 的子群 $H$ 的问题。与传统的假设 $H$ 已知的 $H$ 不变网络不同，本文提出了一种在满足特定条件下发现潜在子群的方法。我们的结果表明，通过学习一个 $S_{n}$ 不变函数和一个线性变换，可以发现 $S_{k} (k \leq n)$ 类型的任意子群。我们还证明了循环子群和二面体子群的类似结论。最后，我们提出了一个可以推广至发现 $S_{n}$ 其他子群的通用定理。通过图像数字求和与对称多项式回归任务的数值实验，我们展示了所得结果的适用性。