In this paper we investigate computational properties of the Diophantine problem for spherical equations in some classes of finite groups. We classify the complexity of different variations of the problem, e.g., when $G$ is fixed and when $G$ is a part of the input. When the group $G$ is constant or given as multiplication table, we show that the problem always can be solved in polynomial time. On the other hand, for the permutation groups $S_n$ (with $n$ part of the input), the problem is NP-complete. The situation for matrix groups is quite involved: while we exhibit sequences of 2-by-2 matrices where the problem is NP-complete, in the full group $GL(2,p)$ ($p$ prime and part of the input) it can be solved in polynomial time. We also find a similar behaviour with subgroups of matrices of arbitrary dimension over a constant ring.

翻译：本文研究了几类有限群中球面方程丢番图问题的计算性质。我们对该问题的不同变体的复杂度进行了分类，例如当$G$固定时以及当$G$作为输入的一部分时。当群$G$为常数或以乘法表给出时，我们证明该问题总能在多项式时间内求解。另一方面，对于置换群$S_n$（其中$n$为输入的一部分），该问题是NP完全的。矩阵群的情况则较为复杂：虽然我们构造了2×2矩阵序列使该问题为NP完全的，但在完整群$GL(2,p)$（$p$为素数且为输入的一部分）中，该问题可在多项式时间内求解。我们还发现常数环上任意维数矩阵的子群具有类似性质。