A340521 List of possible orders of automorphism groups of finite groups.

1, 2, 4, 6, 8, 10, 12, 16, 18, 20, 22, 24, 28, 30, 32, 36, 40, 42, 44, 46, 48, 52, 54, 56, 58, 60

Offset: 1

Author

Des MacHale, Feb 05 2021

Keywords

• nonn
• more

Comments

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.

This is a supersequence of A002202 since |Aut(Z/nZ)| = phi(n). - Jianing Song, Feb 05 2021

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

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

Links

• D. MacHale and R. Sheehy, Finite groups with few automorphisms, Mathematical Proceedings of the Royal Irish Academy, Vol. 104A, No. 2 (December 2004), 231-238.

Crossrefs

Cf. A002174, A002202, A049225, A137315.