A study on Ivar(G) of a p-group fixing certain subgroups
محورهای موضوعی : Group theory
S. Barin
1
,
M. M. Nasrabadi
2
1 - Department of Mathematics, University of Birjand, Birjand, Iran
2 - Department of Mathematics, University of Birjand, Birjand, Iran
کلید واژه: IA-group, IA-central subgroup, Ivar(G), normal subgroups,
چکیده مقاله :
Let $G$ be a group, and $M$ and $N$ be two normal subgroups of $G$. In this paper, we first introduce a subgroup $\mathcal{E}(G)$ and consider the set of all automorphisms of $G$ which centralize $G/M$ and $N$. Then we investigate the conditions in which this set of automorphisms with different $M$ and $N$ is equal or to be equal with $Ivar(G)$.
Let $G$ be a group, and $M$ and $N$ be two normal subgroups of $G$. In this paper, we first introduce a subgroup $\mathcal{E}(G)$ and consider the set of all automorphisms of $G$ which centralize $G/M$ and $N$. Then we investigate the conditions in which this set of automorphisms with different $M$ and $N$ is equal or to be equal with $Ivar(G)$.
[1] M. S. Attar, Semicomplete finite p-groups, Algeb. Colloq. 18 (2011), 937-944.
[2] Z. Azhdari, M. Akhavan-Malayeri, On inner automorphisms and central automorphisms of nilpotent group of class 2, J. Alge. Appl. 10 (4) (2011), 1283-1290.
[3] S. Bachmuth, Automorphisms of free metabelian groups, Trans. Amer. Math. Soc. 118 (1965), 93-104.
[4] S. Barin, M. M. Nasrabadi, Two new characteristic subgroups, J. Linear. Topological. Algebre. 12 (4) (2023), 273-279.
[5] M. J. Curran, D. J. McCaughan, Central automorphisms that are almost inner, Commun. Algebra. 29(b) (2001), 2081-2087.
[6] R. G. Ghumde, Solvability and polycyclicity of group of IA-automorphisms, J. Alge. Numb. Theo. Appl. 48 (2) (2020), 203-209.
[7] R. G. Ghumde, S. H. Ghate, Bounds of the group of IA-automorphisms, Eurasian Math. J. 7 (4) (2016), 30-45.
[8] R. G. Ghumde, S. H. Ghate, IA-automorphisms of p-groups, finite polycyclic groups and other results, Mat. Ves. 67 (2015), 194-200.
[9] S. H. Jafari, Central automorphism groups fixing the center element-wise, Inter. Elect. J. Algebra. 9 (2011), 167-170.
[10] P. K. Rai, On IA-automorphisms that fix the center element-wise, Proc. Math. Sci. 124 (2014), 169-174.
[11] M. K. Yadav, On central automorphisms fixing the center element-wise, Comm. Algebra. 37 (2009), 4325-4331.
