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A341298 Orders of complete groups. 2
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, 1012, 1080 (list; graph; refs; listen; history; text; internal format)
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 788953370457 = 3*19*7^12. [Corrected by Jianing Song, Aug 25 2023]
Recently, Dark showed that the smallest possible nontrivial complete group G of odd order has order 352947 = 3*7^6. [In fact, for every prime p == 1 (mod 3), there exists a complete group of order 3*p^6, and it occurs as the automorphism group of a group of order 3*p^5. This means that there are infinitely many odd terms in this sequence. See the M. John Curran and Rex S. Dark link. - Jianing Song, Aug 25 2023]
From Jianing Song, Aug 25 2023: (Start)
The holomorph (see the Wikipedia link) of an abelian group of odd order is a complete group. See Theorem 3.2, Page 618 of the W. Peremans link.
No prime power (A246655) is a term. See the first Groupprops link.
The automorphism group of a complete group is isomorphic to itself. The converse is not true, as shown by the counterexamples D_8 and D_12. In contrast with the fact that the holomorph of a complete group is isomorphic to the external direct product of two copies of it (see the second Groupprops link), the holomorph of D_8 (SmallGroup(64,134)) is not isomorphic to D_8 X D_8 = SmallGroup(64,226), and the holomorph of D_12 (SmallGroup(144,154)) is not isomorphic to D_12 X D_12 = SmallGroup(144,192). (End)
M. John Curran and Rex S. Dark, Complete Groups of Order 3p^6, Advances in Group Theory and Applications, 2 (2016), pp. 1-12.
R. S. Dark, A complete group of odd order, Mathematical Proc. Cambridge Philosophical Society, Vol. 77, No. 1, January 1975, pp. 21-28.
W. Peremans, Completeness of Holomorphs, Nederl. Akad. Wetensch. Proc. Ser. A, 60. (1957) 608-619.
Wikipedia, Holomorph
a(3) = 20 because 20 is the third number for which there is a complete group of that order.
Sequence in context: A020889 A334817 A084682 * A308324 A044970 A345910
Bob Heffernan and Des MacHale, Feb 10 2021
a(36) and a(37) added by Jianing Song, Aug 25 2023

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