In this paper, we study the notion of the twisted conjugacy separability developed in Fel’shtyn and Troitsky (J K-Theory 1:1–40, 2008). We give an example of group with twisted conjugacy separability, but without residual finiteness. Thus, we find the negative answers on the several open questions formulated by Fel’shtyn and Troitsky (Topol Appl 157:1724–1735, 2010). In addition, we discuss the generalization of usual the commutator subgroup, especially for some relations among Nielsen fixed point theory and the twisted commutator subgroup.

中文翻译：

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扭曲的换向器子群以及Fel'shtyn和Troitsky的问题

在本文中，我们研究了Fel'shtyn和Troitsky提出的扭曲共轭可分性的概念（J K-理论1：1–40，2008年）。我们给出了一个具有扭曲共轭可分离性但没有剩余有限性的组的例子。因此，我们在Fel'shtyn和Troitsky提出的几个开放性问题上找到了否定答案（Topol Appl 157：1724-1735，2010）。另外，我们讨论了通常的换向器子群的推广，特别是对于尼尔森不动点理论和扭曲的换向器子群之间的某些关系。