A classical theorem of Reinhold Baer shows that any group which is finite over its k-th centre has a finite (k + 1)-th term of its lower central series. Although the converse statement is false in general, Philip Hall proved that for any group G the finiteness of γk + 1(G) implies that the index $|G:\zeta_{2k}(G)|$ is finite. Similar situations have been investigated when finiteness is replaced by suitable weaker conditions. Moreover, it was proved by Yuriĭ Merzljakov that Baer’s theorem and its potential converse do hold for linear groups. The aim of this paper is to obtain results of the latter type for several other finiteness conditions. Finally, although a result of Baer type does not hold for the class of soluble-by-finite reduced minimax groups, we prove that for this class a theorem of Hall type is true in arbitrary groups.

中文翻译：

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线性群中的上下中心级数

Reinhold Baer 的经典定理表明，任何在其第 k 个中心上有限的群都具有其下中心级数的有限 (k + 1) 项。尽管相反的陈述通常是错误的，但 Philip Hall 证明了对于任何群 G，γk + 1(G) 的有限性意味着索引 $|G:\zeta_{2k}(G)|$ 是有限的。当有限性被适当的较弱条件代替时，已经研究了类似的情况。此外，Yuriĭ Merzljakov 证明了 Baer 定理及其潜在的逆对确实适用于线性群。本文的目的是在其他几种有限性条件下获得后一种类型的结果。最后，虽然 Baer 型的结果不适用于可溶的有限化约极小极大群类，但我们证明了对于此类，霍尔型定理在任意群中都是正​​确的。