We study both the Submonoid Membership problem and the Rational Subset Membership problem in finitely generated nilpotent groups. We give two reductions with important applications. First, Submonoid Membership in any nilpotent group can be reduced to Rational Subset Membership in smaller groups. As a corollary, we prove the existence of a group with decidable Submonoid Membership and undecidable Rational Subset Membership, confirming a conjecture of Lohrey and Steinberg. Second, the Rational Subset Membership problem in $H_3(\mathbb Z)$ can be reduced to the Knapsack problem in the same group, and is therefore decidable. Combining both results, we deduce that the filiform $3$-step nilpotent group has decidable Submonoid Membership.

翻译：我们研究有限生成幂零群中的子幺半群成员问题和有理子集成员问题。我们给出了两个具有重要应用的约简。首先，任何幂零群中的子幺半群成员问题都可以约简为更小群中的有理子集成员问题。作为推论，我们证明存在一个群，其子幺半群成员问题可判定而有理子集成员问题不可判定，验证了Lohrey和Steinberg的一个猜想。其次，$H_3(\mathbb Z)$中的有理子集成员问题可以约简为同一群中的背包问题，因此是可判定的。结合这两个结果，我们推导出纤维型3步幂零群具有可判定的子幺半群成员问题。