Finite groups whose automizers of all abelian subgroups are either small or large

Abstract

The automizer of a subgroup H of a group G is Aut${}_G(H)=N_G(H)/C_G(H)$. Aut_G(H) can be regarded as a subgroup of Aut(H) and Aut_G(H) contains an isomorphic copy of Inn(H), we shall say that Aut_G(H) is large if Aut${}_G(H)\cong Aut(H)$ and small if Aut${}_G(H)\cong$ Inn(H). In this paper, we study the finite groups G satisfying the condition: Aut${}_G(H)\cong$ Aut(H) or Aut${}_G(H)\cong$ Inn(H) for every abelian subgroup H of G.

Keywords:

• Abelian subgroups
• automizers
• solvable groups

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 20D10
• 20D20

Acknowledgements

The authors are grateful to the referee, who provided her/his valuable suggestions and detailed reports.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

W. Meng is supported by National Natural Science Foundation of China (11761079); H. L. Yao is supported by National Natural Science Foundation of China (11671126); J. K. Lu is supported by National Natural Science Foundation of China (11861015) and Guangxi Natural Science Foundation Program (2016GXNSFAA380156).