OFFSET
1,2
COMMENTS
The sum of the orders of all elements of a finite group G is denoted by psi(G).
psi(A X B) = psi(A)*psi(B) for finite groups A and B of coprime orders.
psi(G) <= 7/11 psi(C_n) < psi(C_n) for every noncyclic finite group G of order n.
psi(G) < 1/(p - 1) psi(C_n) for every noncyclic finite group G of order n, where p the smallest prime divisor of n.
Conjecture: If S is a simple group and G is a soluble group satisfying |S|=|G|, then psi(S) < psi(G).
LINKS
M. Farrokhi D. G., Table of n, a(n) for n = 1..1027
H. Amiri, S. M. Jafarian Amiri, and I. M. Isaacs, Sums of element orders in finite groups, Comm. Algebra 37(9) (2009), 2978-2980.
M. Herzog, P. Longobardi, and M. Maj, Properties of finite and periodic groups determined by their element of orders (a survey), Group theory and computation, 59-90, Indian Stat. Inst. Ser., Springer, Singapore, 2018.
EXAMPLE
psi(C_6) = 1 + 2 + 3 + 3 + 6 + 6 = 21.
PROG
(GAP) Sum(List(G, Order));
CROSSREFS
KEYWORD
nonn
AUTHOR
M. Farrokhi D. G., Sep 16 2020
STATUS
approved