Some Properties on Sums of Element Orders in Finite p-groups
In literature, there are many papers on the sum of element orders of a finite group. In this study, in particular, we deal with thecases 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 suchsums. 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.
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- [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.