A337706 Possible values for the product of the orders of all elements of a finite group.

1, 2, 8, 9, 32, 72, 128, 512, 625, 648, 2048, 6561, 8192, 20000, 32768, 41472, 52488, 117649, 131072, 524288, 2654208, 3359232, 4782969, 8388608, 12500000, 15059072, 33554432, 134217728, 214990848, 536870912, 2147483648, 2448880128

Offset: 1

Author

M. Farrokhi D. G., Sep 16 2020

Keywords

• nonn

Comments

The product of the orders of all elements of a finite group G is denoted by P(G).

P(A X B) = P(A)^|B|P(B)^|A| for finite groups A and B of coprime orders.

P(G) < P(C_n) for every noncyclic finite group G of order n.

Examples

			P(C_6) = 1*2*3*3*6*6 = 648.
		

Links

• M. Farrokhi D. G., Table of n, a(n) for n = 1..2733
• M. Garonzi and M. Patassini, Inequalities detecting structural properties of a finite group, Comm. Algebra (2017) 45(2), 677-687.
• 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

Product(List(G, Order))