TY - JOUR
T1 - An exact upper bound for sums of element orders in non-cyclic finite groups
AU - Herzog, Marcel
AU - Longobardi, Patrizia
AU - Maj, Mercede
N1 - Publisher Copyright:
© 2017 Elsevier B.V.
PY - 2018/7
Y1 - 2018/7
N2 - Denote the sum of element orders in a finite group G by ψ(G) and let Cn denote the cyclic group of order n. Suppose that G is a non-cyclic finite group of order n and q is the least prime divisor of n. We proved that ψ(G)≤[Formula presented]ψ(Cn) and ψ(G)<[Formula presented]ψ(Cn). The first result is best possible, since for each n=4k, k odd, there exists a group G of order n satisfying ψ(G)=[Formula presented]ψ(Cn) and the second result implies that if G is of odd order, then ψ(G)<[Formula presented]ψ(Cn). Our results improve the inequality ψ(G)<ψ(Cn) obtained by H. Amiri, S.M. Jafarian Amiri and I.M. Isaacs in 2009, as well as other results obtained by S.M. Jafarian Amiri and M. Amiri in 2014 and by R. Shen, G. Chen and C. Wu in 2015. Furthermore, we obtained some ψ(G)-based sufficient conditions for the solvability of G.
AB - Denote the sum of element orders in a finite group G by ψ(G) and let Cn denote the cyclic group of order n. Suppose that G is a non-cyclic finite group of order n and q is the least prime divisor of n. We proved that ψ(G)≤[Formula presented]ψ(Cn) and ψ(G)<[Formula presented]ψ(Cn). The first result is best possible, since for each n=4k, k odd, there exists a group G of order n satisfying ψ(G)=[Formula presented]ψ(Cn) and the second result implies that if G is of odd order, then ψ(G)<[Formula presented]ψ(Cn). Our results improve the inequality ψ(G)<ψ(Cn) obtained by H. Amiri, S.M. Jafarian Amiri and I.M. Isaacs in 2009, as well as other results obtained by S.M. Jafarian Amiri and M. Amiri in 2014 and by R. Shen, G. Chen and C. Wu in 2015. Furthermore, we obtained some ψ(G)-based sufficient conditions for the solvability of G.
UR - http://www.scopus.com/inward/record.url?scp=85025643336&partnerID=8YFLogxK
U2 - 10.1016/j.jpaa.2017.07.015
DO - 10.1016/j.jpaa.2017.07.015
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AN - SCOPUS:85025643336
SN - 0022-4049
VL - 222
SP - 1628
EP - 1642
JO - Journal of Pure and Applied Algebra
JF - Journal of Pure and Applied Algebra
IS - 7
ER -