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A341298 Orders of complete groups. 0
1, 6, 20, 24, 42, 54, 110, 120, 144, 156, 168, 216, 252, 272, 320, 324, 336, 342, 384, 432, 480, 486, 500, 506, 660, 720, 800, 812, 840, 864, 930, 936, 960, 972, 1008 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

A finite group G is called complete if Aut G = Inn G and Z(G) = {1} i.e. G has no outer automorphisms and the center of G is trivial.

The symmetric group S(n) of order n! is complete for n not equal to 2 or 6.

If p is an odd prime, there is a complete group of order p(p-1) and a complete group of order p^m*(p^m - p^(m-1)) for each m.

Dark in 1975 discovered a nontrivial complete group G of odd order. It has order 33209467522096377 = 3*19*17^12.

Recently, Dark showed that the smallest possible nontrivial complete group G of odd order has order 352947 = 3*7^6.

LINKS

Table of n, a(n) for n=1..35.

R. S. Dark, A complete group of odd order, Mathematical Proc. Cambridge Philosophical Society, Vol. 77, No. 1, January 1975, pp. 21-28.

EXAMPLE

a(3) = 20 because 20 is the third number for which there is a complete group of that order.

CROSSREFS

Sequence in context: A020889 A334817 A084682 * A308324 A044970 A345910

Adjacent sequences:  A341295 A341296 A341297 * A341299 A341300 A341301

KEYWORD

nonn,more

AUTHOR

Bob Heffernan and Des MacHale, Feb 10 2021

STATUS

approved

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Last modified September 19 06:12 EDT 2021. Contains 347551 sequences. (Running on oeis4.)