A341298 Orders of complete groups.

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

Offset: 1

Author

Bob Heffernan and Des MacHale, Feb 10 2021

Keywords

• nonn
• more

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 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)

Examples

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

Links

• 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.
• Groupprops, Group isomorphic to its automorphism group
• Groupprops, Holomorph of a group
• W. Peremans, Completeness of Holomorphs, Nederl. Akad. Wetensch. Proc. Ser. A, 60. (1957) 608-619.
• Jianing Song, List of complete groups with order <= 1080
• Wikipedia, Holomorph

Extensions

a(36) and a(37) added by Jianing Song, Aug 25 2023