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A337706 Possible values for the product of the orders of all elements of a finite group. 1
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

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.

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.

EXAMPLE

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

PROG

(GAP) Product(List(G, Order))

CROSSREFS

Sequence in context: A166968 A075644 A088825 * A181887 A221049 A181476

Adjacent sequences:  A337703 A337704 A337705 * A337707 A337708 A337709

KEYWORD

nonn

AUTHOR

M. Farrokhi D. G., Sep 16 2020

STATUS

approved

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Last modified June 17 13:12 EDT 2021. Contains 345080 sequences. (Running on oeis4.)