A337705 Possible sums of orders of elements of finite groups.

1, 3, 7, 11, 13, 15, 19, 21, 23, 25, 27, 31, 33, 39, 43, 45, 47, 49, 55, 57, 59, 61, 63, 67, 71, 73, 75, 77, 79, 83, 85, 87, 95, 97, 99, 101, 103, 105, 111, 113, 115, 119, 121, 125, 127, 129, 133, 135, 143, 147, 151, 153, 157, 159, 161, 163

Offset: 1

Author

M. Farrokhi D. G., Sep 16 2020

Keywords

• nonn

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).

Examples

			psi(C_6) = 1 + 2 + 3 + 3 + 6 + 6 = 21.
		

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.

Programs

GAP

Sum(List(G, Order));