We provide a polynomial time algorithm for deciding the equation solvability problem over finite groups that are semidirect products of a [Formula: see text]-group and an Abelian group. As a consequence, we obtain a polynomial time algorithm for deciding the equivalence problem over semidirect products of a finite nilpotent group and a finite Abelian group. The key ingredient of the proof is to represent group expressions using a special polycyclic presentation of these finite solvable groups.

中文翻译：

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有限群上方程可解性和等价问题的复杂性

我们提供了一种多项式时间算法，用于确定有限群上的方程可解性问题，这些有限群是 [公式：参见文本]-群和阿贝尔群的半直积。因此，我们获得了一个多项式时间算法，用于确定有限幂零群和有限阿贝尔群的半直积上的等价问题。证明的关键成分是使用这些有限可解群的特殊多环表示来表示群表达式。