The structure of finite groups whose elements outside a normal subgroup have prime power orders

The structure of groups in which every element has prime power order (CP-groups) is extensively studied. We first investigate the properties of group G such that each element of G N has prime power order. It is proved that N is solvable or every non-solvable chief factor $H/K$ of G satisfying H≤ N is isomorphic to PSL₂(3ᶠ) with f a 2-power. This partially answers the question proposed by Lewis in 2023, asking whether G M₁₀ ? Furthermore, we prove that if each element x∈ G N has prime power order and CG(x) is maximal in G , then N is solvable. Relying on this, we give the structure of group G with normal subgroup N such that CG(x) is maximal in G for any element x∈ G N . Finally, we investigate the structure of a normal subgroup N when the centralizer CG(x) is maximal in G for any element x∈ N Z(N) , which is a generalization of results of Zhao, Chen, and Guo in 2020, investigating a special case that $N=G$ for our main result. We also provide a new proof for Zhao, Chen, and Guo's results above.