Parisa SEIFIZADEH, AmirAli FAROKHNIAEE, Ali GHOLAMIAN, Mohammad Mehdi NASRABADI

On the autocentralizer subgroups of finite p–groups

ISSN: 1300-0098
Yayın Aralığı: Yılda 6 Sayı
Yayıncı: TÜBİTAK

Let G be a finite group and Aut(G) be the group of automorphisms of G. Then, the autocentralizer of an automorphism α ∈ Aut(G) in G is defined as CG(α) = {g ∈ G|α(g) = g}. Let Acent(G) = {CG(α)|α ∈ Aut(G)}. If |Acent(G)| = n, then G is an n–autocentralizer group. In this paper, we classify all n–autocentralizer abelian groups for n = 6, 7 and 8. We also obtain a lower bound on the number of autocentralizer subgroups for p–groups, where p is a prime number. We show that if p ̸= 2, there is no n–autocentralizer p–group for n = 6, 7. Moreover, if p = 2, then there is no 6–autocentralizer p–group.

PDF

___

[1] Ashrafi AR. On finite groups with a given number of centralizers. Algebra Colloquium 2000; 7 (2): 139-146. doi: 10.1007/s10011-000-0139-5
[2] Bacon MR, Kappe LC. On capable p–groups of nilpotentcy class two. Illinois Journal of Mathematics 2003; 47 (1/2): 49-62. doi: 10.1215/ijm/1258488137
[3] Belcastro SM, Sherman GJ. Counting centralizers in finite groups. Mathematical Sciences Technical Reports 1994; 67 (5): 366-374. doi: 10.2307/2690998
[4] Magidin A. Capable 2–generator 2–groups of class two. Communications in Algebra 2006; 34 (6): 2183-2193. doi: 10.1080/00927870600549717
[5] Nasrabadi MM, Gholamian A. On A–nilpotent abelian groups. Proceedings Indian Academy of Sciences 2014; 124 (4): 517-525. doi: 10.1007/s12044-014-0197-0
[6] Nasrabadi MM, Gholamian A. On finite n–Acentralizer groups. Communications in Algebra 2015; 43 (2): 378-383. doi: 10.1080/00927872.2013.842244
[7] Robinson DJS. Course in the Theory of Groups. New York, NY, USA: Springer-Verlag, 1980.
[8] Sarmin NH, Barakat Y. Specific automorphisms on a 2–generated p-group of class two. AIP International Conference on Mathematical Seciences and Statistics 2013; 1557 ( 41): 35-37. doi: 10.1063/1.4823871