Groups with some central automorphisms fixing the central kernel quotient

Document Type : Research Paper

Author

Department of Mathematics, Payame Noor University (PNU), Iran

Abstract

Let $G$ be a group. An automorphism $\alpha$ of a group $G$ is called a central automorphism, if $x^{-1}x^{\alpha}\in Z(G)$ for all $x\in G$. Let $L_c(G)$ be the central kernel of $G$, that is the set of elements of $G$ fixed by all central  automorphisms of $G$ and $Aut_{L_c}(G)$ denote the group of all central automorphisms of $G$ fixing $G/L_c(G)$ element-wise. In the present paper, we investigate the properties of such automorphisms. Moreover, a full classification of $p$-groups $G$ of order at most $p^5$ where $Aut_{L_c}(G)=Inn(G)$ is also given.

Keywords

References

[1] M. J. Curran, D. J. McCaughan, Central automorphisms that are almost inner, Comm. Algebra vol. 29, no. 5 (2001) 2081{2087.
[2] S. Davoudirad, M. R. R. Moghaddam, M. A. Rostamyari, Some properties of central kernel and central autocommutator subgroups, J. Algebra Appl vol. 15, no. 7 (2016) 16501281{7.
[3] S. Davoudirad, M. R. R. Moghaddam, M. A. Rostamyari, Some properties of central kernel quotient of a group, Asian-European J. Math. vol. 1 (2020) 1{9.
[4] The GAP Group, GAP-Groups, Algorithms and Programing, Version 4.11.1; 2021, (http://www.gap-system.org).
[5] F. Haimo, Normal automorphisms and their  xed points, Trans. Amer. Math. Soc. vol. 78, no. 1 (1955) 150-167.
[6] P. Hall, The classi cation of prime power groups, J. Reine Angew. Math. vol. 182 (1940) 130{141.
[7] P. V. Hegarty, The absolute centre of a group, J. Algebra vol. 169 (1994) 929{935.
[8] R. James, The groups of order p6 (p an odd prime), Math. Comp. vol. 34 (1980) 613{637.
[9] H. Meng, X. Guo, The absolute center of  nite groups, J. Group Theory vol. 18 (2015) 887{904.
[10] M. Morigi, On the minimal number of generators of  nite non-abelian p-groups having an abelian automorphism group, Comm. Algebra vol. 23, no. 6 (1995) 2045{2065.
[11] L. Redei, Endliche p-Gruppen, Akademiai Kiado, Budapest, 1989.
[12] R. Soleimani, On some p-subgroups of automorphism group of a  nite p-group, Vietnam J. Math. vol. 36, no. 1 (2008) 63{69.
[13] D. L. Winter, The automorphism group of an extra-special p-group, Rocky Mountain J. Math. vol. 2, no. 2 (1972) 159{168.

Files

History

  • Receive Date: 16 March 2022
  • Revise Date: 26 July 2022
  • Accept Date: 17 September 2022

Share

How to cite

Statistics