A341824 Number of groups of order 2^n which occur as Aut(G) for some finite group G.

1, 1, 2, 3, 4, 9, 14, 33

Offset

0,3

Comments

The number of groups of order 2^n is A000679(n); the percentage of the 2-groups which occur as automorphism groups appears to decrease as n increases: 100, 100, 100, 60, 28.5, 17.6, 5.2, 1.4, ...

Jianing Song remarks that it is also interesting to consider infinite groups, and asks if there is an infinite group G with Aut(G) isomorphic to C_8. Des MacHale, Mar 03 2021, replies that at present this is not known. [Comment edited by N. J. A. Sloane, Mar 07 2021]

Links

Table of n, a(n) for n=0..7.

J. Flynn, D. MacHale, E. A. O'Brien and R. Sheehy, Finite Groups whose Automorphism Groups are 2-groups, Proc. Royal Irish Academy, 94A, (2) 1994, 137-145.

Formula

a(n) <= A000679(n). - Des MacHale, Mar 02 2021

Example

a(5) = 9 because there are nine groups of order 32 which occur as automorphism groups of finite groups.
From Bernard Schott, Feb 26 2021: (Start)
Aut(C_15) = Aut(C_16) = Aut(C_20) = Aut(C_30) ~~ C_4 x C_2 where ~~ stands for "isomorphic to".
Aut(C_4 x C_2) = Aut(D_4) ~~ D_4 where D_4 is the dihedral group of the square.
Aut(C_24) ~~ C_2 x C_2 x C_2 = (C_2)^3.
There exist five groups of order 8 (A054397), the three groups C_4 x C_2, D_4, C_2 x C_2 x C_2 occur as automorphim groups of order 8, but the cyclic group C_8 and the quaternions group Q_8 never occur as Aut(G) for some finite G, so a(3) = 3. (End)

Crossrefs

Cf. A000679, A054397, A340521, A341823, A341825.

Sequence in context: A077906 A342532 A133993 * A122974 A032982 A288856

Adjacent sequences: A341821 A341822 A341823 * A341825 A341826 A341827

Keyword

nonn,more

Author

Des MacHale, Feb 26 2021

Extensions

Offset modified by Jianing Song, Mar 02 2021

Status

approved