login
List of possible orders of automorphism groups of finite groups.
5

%I #23 Feb 10 2021 13:23:54

%S 1,2,4,6,8,10,12,16,18,20,22,24,28,30,32,36,40,42,44,46,48,52,54,56,

%T 58,60

%N List of possible orders of automorphism groups of finite groups.

%C The terms shown here match the initial terms of all of A002174, A002202, A049225, but this is a strictly different sequence since it is known that it contains 3^7 = 2187 (which is the smallest odd term greater than 1), whereas for the other three sequences all terms greater than 1 are even.

%C This is a supersequence of A002202 since |Aut(Z/nZ)| = phi(n). - _Jianing Song_, Feb 05 2021

%C John Bray has produced a group G of order 3^2*19 = 171 such that |Aut G| = 1026 = 2.3^3.19. So 1026 is in the present sequence but is not in A002202. So the present sequence contains both odd and even terms not in A002202. - _Des MacHale_, Feb 10 2021

%C For more about this problem, see the references in A137315.

%H D. MacHale and R. Sheehy, <a href="http://www.jstor.org/stable/40656888">Finite groups with few automorphisms</a>, Mathematical Proceedings of the Royal Irish Academy, Vol. 104A, No. 2 (December 2004), 231-238.

%Y Cf. A002174, A002202, A049225, A137315.

%K nonn,more

%O 1,2

%A _Des MacHale_, Feb 05 2021