Abstract
The intersection of particular subgroups is a kind of interesting substructure in group theory. Let G be a finite group and D(G) be the intersection of the normalizers of the derived subgroups of all the subgroups of G. A group G is called a D-group if G = D(G). In this paper, we determine the nilpotency class of the nilpotent residual \({G^{\cal N}}\) and investigate the structure of D(G) by a new concept called the IO-D-group. A non-D-group G is called an IO-D-group (inner-outer-D-group) if all of its proper subgroups and proper quotient groups are D-groups. The structure of IO-D-groups are described in detail in this paper. As an application of the classification of IO-D-groups, we prove that G is a D-group if and only if any subgroup of G generated by 3 elements is a D-group.
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Acknowledgements
This work was supported by National Natural Science Foundation of China (Grant Nos. 11631001 and 12071181). The authors are grateful to Professors Bertram A. F. Wehrfritz and Yanjun Liu who provided profound suggestions. The authors also thank the referees who provided detailed reports and improved the formulation of Definition 3.6 and Theorem 3.7.
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Shen, Z., Li, S. & Zhang, J. Derived norms of finite groups. Sci. China Math. 65, 2493–2502 (2022). https://doi.org/10.1007/s11425-021-1942-9
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DOI: https://doi.org/10.1007/s11425-021-1942-9