Some Properties on Sums of Element Orders in Finite p-groups

Year 2020, Volume: 8 Issue: 1, 51 - 54, 20.03.2020

Nil Mansuroğlu

https://doi.org/10.36753/mathenot.616086

Abstract

In literature, there are many papers on the sum of element orders of a finite group. In this study, in particular, we deal with the
cases in finite p-groups. Our main aim is to investigate the sums of element orders in finite p-groups and to give some properties of such
sums. Let (G) denote the sum of element orders of a finite group G. As an immediate consequence, we proved that \psi(G) ≤ 3 /4 \psi(C) and
\psi(G) < 1/p-1\psi (C), where G is a non-cyclic finite p-group of order p^r and C is a cyclic group of order p^r for some prime p.

Keywords

finite p-group, group element order;, sum of element orders

References

• [1] H. Amiri, S.M. Jafarian Amiri and I.M. Isaacs, Sums of element ordersin finite groups, Comm. Algebra. Volume 37, 2978-2980, 2009.
• [2] H. Amiri, S.M. Jafarian Amiri, Sum of element orders on finite groupsof the same order, J. Algebra Appl., 10 (2), 187-190, 2011.
• [3] S.M. Jafarian Amiri, Second maximum sum of element orders on finitenilpotent groups, Comm. Algebra, 41 (6), 2055-2059, 2013.
• [4] H. Amiri, S.M. Jafarian Amiri, Second maximum sum of element orderson finite groups, J. Pure Appl. Algebra, 218 (3), 531-539, 2014.
• [5] M. Herzog, P. Longobardi and M. Maj, An exact upper bound for sumsof element orders in non-cyclic finite groups, J. Pure and Applied Algebra, 222 (7), 1628{1642, 2018.
• [6] E.I. Khukhro, p-Automorphisms of Finite p-Groups, London Mathematical Society Lecture Note Series 246, Cambridge University Press,1997.
• [7] N. Mansuro˘glu, Sums of element orders in symmetric groups, AfyonKocatepe University Journal of Science and Engineering, 18, 884-887,2018.
• [8] I.I. Martin, Finite group theory, Graduate Studies in Mathematics, 92,American Mathematical Society, Providence, RI, 2008.
• [9] D. J. S. Robinson, A Course in the Theory of Groups, 2nd edition,Springer-Verlag, New York, 1996.
• [10] R. Shen, G. Chen and C. Wu, On groups with the Second Largest Valueof the Sum of Element Orders, Comm. Algebra, 43 (6), 2618-2631, 2015.