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Orders of complete groups.
3

%I #55 Aug 25 2023 20:02:14

%S 1,6,20,24,42,54,110,120,144,156,168,216,252,272,320,324,336,342,384,

%T 432,480,486,500,506,660,720,800,812,840,864,930,936,960,972,1008,

%U 1012,1080

%N Orders of complete groups.

%C 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.

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

%C 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.

%C 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]

%C 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]

%C From _Jianing Song_, Aug 25 2023: (Start)

%C 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.

%C No prime power (A246655) is a term. See the first Groupprops link.

%C 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)

%H M. John Curran and Rex S. Dark, <a href="https://www.advgrouptheory.com/journal/Volumes/2/M.J.%20Curran,%20R.S.%20Dark%20-%20Complete%20groups%20of%20order%203p6.pdf">Complete Groups of Order 3p^6</a>, Advances in Group Theory and Applications, 2 (2016), pp. 1-12.

%H R. S. Dark, <a href="https://doi.org/10.1017/S0305004100049392">A complete group of odd order</a>, Mathematical Proc. Cambridge Philosophical Society, Vol. 77, No. 1, January 1975, pp. 21-28.

%H Groupprops, <a href="https://groupprops.subwiki.org/wiki/Group_isomorphic_to_its_automorphism_group">Group isomorphic to its automorphism group</a>

%H Groupprops, <a href="https://groupprops.subwiki.org/w/index.php?title=Holomorph_of_a_group">Holomorph of a group</a>

%H W. Peremans, <a href="https://core.ac.uk/download/pdf/82560418.pdf">Completeness of Holomorphs</a>, Nederl. Akad. Wetensch. Proc. Ser. A, 60. (1957) 608-619.

%H Jianing Song, <a href="/A341298/a341298_1.txt">List of complete groups with order <= 1080</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Holomorph_(mathematics)">Holomorph</a>

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

%K nonn,more

%O 1,2

%A _Bob Heffernan_ and _Des MacHale_, Feb 10 2021

%E a(36) and a(37) added by _Jianing Song_, Aug 25 2023