设$G$和$H$是相互兼容的组。我们用$[G,H]$表示$G$在$H$下的导子群。证明了如果集合$\{g^{-1}g^h \mid g \in G, h \in H\}$有$m$元素，则其导数$[G,H]$是有限的，且具有$m$ -有界阶.进一步证明了如果所有张量的集合$T_{\otimes}(G,H) = \{g\otimes h \mid g \in G, h\in H\}$都有$m$元，则非阿贝尔张量积$G \otimes H$是有限的，且具有$m$ -有界阶.我们还研究了群的非交换张量平方的有限性条件。
英文摘要：
Let $G$ and $H$ be groups that act compatibly on each other. We denote by $[G,H]$ the derivative subgroup of $G$ under $H$. We prove that if the set $\{g^{-1}g^h \mid g \in G, h \in H\}$ has $m$ elements, then the derivative $[G,H]$ is finite with $m$-bounded order. Moreover, we show that if the set of all tensors $T_{\otimes}(G,H) = \{g\otimes h \mid g \in G, h\in H\}$ has $m$ elements, then the non-abelian tensor product $G \otimes H$ is finite with $m$-bounded order. We also examine some finiteness conditions for the non-abelian tensor square of groups.
Schur's theorem and its relation to the closure properties of the non-abelian tensor product
Universidade de Brasilia（巴西利亚大学）；Vladimir Chavchanidze Institute of Cybernetics of the Georgian Technical University（格鲁吉亚理工大学弗拉基米尔·查夫奇尼泽网络学研究所）；University of Santiago de Compostela（圣地亚哥德孔波斯特拉大学）
On the residual finiteness of the non-abelian tensor square $G \otimes G$, the non-abelian exterior square $G \wedge G$ and the weak commutativity construction $\X(G)$
Intrinsic tensor products and a Ganea-type extension of the five-term exact sequence
Universidade de Coimbra（科英布拉大学）；Institut de Recherche en Mathématique et Physique（数学与物理研究所）；Max-Planck-Institut für Mathematik（马克斯·普朗克数学研究所）；Chern Institute of Mathematics（陈省身数学研究所）；CINVESTAV（墨西哥国家科学技术研究中心）；North East Normal University（东北师范大学）；Fonds de la Recherche Scientifique–FNRS（科学资助基金-比利时国家科学基金会）；Fundação para a Ciência e a Tecnologia（科学与技术基金会）；York University（约克大学）；Universidad de Vigo（维戈大学）
