Finite groups in which every self-centralizing subgroup is a TI-subgroup or subnormal or has $p^{\mathrm{\prime }}$-order

We first give complete characterizations of the structure of finite group $G$ in which every subgroup (or non-nilpotent subgroup, or non-abelian subgroup) is a TI-subgroup or subnormal or has $p^{\mathrm{\prime }}$-order for a fixed prime divisor $p$ of $\mathrm{\mid }G\mathrm{\mid }$. Furthermore, we prove that every self-centralizing subgroup (or non-nilpotent subgroup, or non-abelian subgroup) of $G$ is a TI-subgroup or subnormal or has $p^{\mathrm{\prime }}$-order for a fixed prime divisor $p$ of $\mathrm{\mid }G\mathrm{\mid }$ if and only if every subgroup (or non-nilpotent subgroup, or non-abelian subgroup) of $G$ is a TI-subgroup or subnormal or has $p^{\mathrm{\prime }}$-order. Based on these results, we obtain the structure of finite group $G$ in which every self-centralizing subgroup (or non-nilpotent subgroup, or non-abelian subgroup) is a TI-subgroup or subnormal or has $p^{\mathrm{\prime }}$-order for a fixed prime divisor $p$ of $\mathrm{\mid }G\mathrm{\mid }$.