Abstract
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.
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