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

1,2

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

Sequence in context: A045062 A111068 A322430 * A102213 A276283 A158942

Adjacent sequences: A337702 A337703 A337704 * A337706 A337707 A337708

Keyword

nonn

Author

M. Farrokhi D. G., Sep 16 2020

Status

approved