Baer定理主题的新变体
New variations on the theme of Baer's theorem
AI总结：
本文证明，对于广义可解群，若下中心级数第s项与上中心级数第t项的交的商群具有有限秩r，则第s+t项下中心群具有有限且受(r,s,t)界定的秩；类似地，将有限秩条件替换为有界大小的Chernikov群也得到相应结果，推广了经典Baer定理。
摘要：
设$\gamma_s(G)$和$Z_s(G)$分别表示群$G$的下中心级数和上中心级数的第$s$项。R. Baer的一个经典定理指出，如果$Z_s(G)$在$G$中具有有限指数$n$，那么$\gamma_{s+1}(G)$也是有限的。在本文中，我们证明，如果$G$是一个广义可解群，使得对于某些$s,t$，商群$\gamma_s(G)/(\gamma_s(G) \cap Z_t(G))$具有有限秩$r$，那么$\gamma_{s+t}(G)$的秩是有限的，并且受$(r,s,t)$界定。此外，还将有限秩假设替换为有界大小的Chernikov群条件，得到了相应的结果。这些结果推广了经典Baer定理的最新推广形式。
英文摘要：
Let $γ_s(G)$ and $Z_s(G)$ denote the $s$-th terms of the lower and upper central series of a group $G$, respectively. A classical theorem by R. Baer states that if $Z_s(G)$ has finite index $n$ in $G$, then $γ_{s+1}(G)$ is also finite. In this paper, we prove that if $G$ is a generalized soluble group such that the quotient $γ_s(G)/(γ_s(G) \cap Z_t(G))$ has finite rank $r$ for some $s,t$, then the rank of $γ_{s+t}(G)$ is finite and $(r,s,t)$-bounded. Moreover, a corresponding result replacing the finite-rank assumption by the condition to be a Chernikov group of bounded size is also obtained. These results extend recent generalizations of the classical Baer's theorem.
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