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Isomorphism classes and zeta-functions of some nilpotent groups. II. (English) Zbl 1345.20030
Let $$G$$ be a finitely generated group. The subgroup zeta function of $$G$$, denoted by $$\zeta_G(s)$$, is defined as the Dirichlet series associated to the sequence $$\{a_n(G)\}_n$$, where $$a_n(G)$$ is the number of subgroups of $$G$$ of index $$n$$. Zeta functions of groups were introduced by F. J. Grunewald, D. Segal and G. C. Smith: they proved in particular that $$\zeta_G(s)$$ has remarkable properties when $$G$$ belongs to the class of torsion-free finitely generated nilpotent groups ($$\mathcal T$$-groups). It is known that for the class of all $$\mathcal T$$-groups, the isomorphim classes are not determined by zeta functions. However this article gives a class of $$\mathcal T$$-groups with the property that if $$G_1$$ and $$G_2$$ belong to this class, then $$\zeta_{G_1}(s)=\zeta_{G_2}(s)$$ if and only if $$G_1$$ and $$G_2$$ are isomorphic.
For part I see [the author, Tokyo J. Math. 36, No. 1, 163-175 (2013; Zbl 1287.20035)].
##### MSC:
 20E07 Subgroup theorems; subgroup growth 20F18 Nilpotent groups 11M41 Other Dirichlet series and zeta functions
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##### References:
 [1] F. J. Grunewald and R. Scharlau, A note on finitely generated torsion-free groups of class 2, J. Algebra 58 (1979), 162-175. · Zbl 0406.20031 [2] F. J. Grunewald, D. Segal and G. C. Smith, Subgroups of finite index in nilpotent groups, Invent. Math. 93 (1988), 185-223. · Zbl 0651.20040 [3] F. Hyodo, Isomorphism classes and Zeta-functions of some Nilpotent Groups, Tokyo J. Math. 36 (2013), 163-175. · Zbl 1287.20035 [4] P. F. Pickel, Finitely generated nilpotent groups with isomorphic finite quotients, Trans. Amer. Math. Soc. 160 (1971), 327-341. · Zbl 0235.20027 [5] C. Voll, Functional equations for zeta functions of groups and rings, Ann. of Math. 172 (2010), no. 2, 1181-1218. · Zbl 1314.11057
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