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.