On the Centralizers of Non-central \(\pi '\)-Elements of a Finite Group

Let G be a group and N be a \(\pi \) $\pi$ -solvable normal subgroup of G with \(\pi \subsetneq \pi (N)\) $\pi ⊊\pi (N)$ , where \(\pi (N)\) $\pi (N)$ is composed by all prime divisors of the order of N. In this paper, we determine the structure of \({N_{\pi '}}{} \textbf{Z}(N)/\textbf{Z}(N)\) $N_{{\pi }^{\prime }}\mathbf{Z}(N)/\mathbf{Z}(N)$ if \(\textbf{C}_G(x)\) ${\mathbf{C}}_G(x)$ is a maximal subgroup of group G for every \(\pi '\) ${\pi }^{\prime }$ -element \(x\in N\setminus \textbf{Z}(N)\) $x\in N\backslash \mathbf{Z}(N)$ , where \(N_{\pi '}\) $N_{{\pi }^{\prime }}$ is a Hall \(\pi '\) ${\pi }^{\prime }$ -subgroup of N. In particular, if \(\pi = \{p\}\) $\pi =\{p\}$ is a set composed by a single prime p, we show that N is solvable, which has its own independent significance. If we assume \(N=G\) $N=G$ in the above results, then it is [8, Theorems A and B] by removing the conditions “G is p-solvable” and “with \(G_{p'}\) $G_{p^{\prime }}$ non-abelian”. We also give a detailed structure description of such groups. Further, we generalize [9, Theorem A] by removing the condition “N is p-solvable”, and also provides a positive answer to [9, Question] by giving the structure of \({N_{p'}}{} \textbf{Z}(N)/\textbf{Z}(N)\) $N_{p^{\prime }}\mathbf{Z}(N)/\mathbf{Z}(N)$ if \(\textbf{C}_G(x)\) ${\mathbf{C}}_G(x)$ is a maximal subgroup of G for every p-regular element \(x\in N{\setminus } \textbf{Z}(N)\) $x\in N\backslash \mathbf{Z}(N)$ .