Let G be a finite group, and H a subgroup of G. We call the subgroup H a TI-subgroup of G if \(H\cap H^{g}=1\) or H for every element g of G. In this paper, we obtain some properties of G when some centralizers of G are TI-groups. Moreover, we also investigate the structure of G under the condition that some centralizers are either TI-subgroups or subnormal subgroups of G, and prove that G is a nilpotent group, or a Frobenius group, or \(G\cong PSL(2,2^n)\) with \(n > 1\) .
