A341823 -id:A341823 - cp's OEIS frontend

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

Des MacHale, Feb 26 2021

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]

			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)

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.

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

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

Offset modified by Jianing Song, Mar 02 2021

A341825 Number of finite groups G with |Aut(G)| = n.

Original entry on oeis.org

2, 3, 0, 4, 0, 6, 0, 7, 0, 2, 0, 9, 0, 0, 0, 11, 0, 4, 0, 7, 0, 2, 0, 22, 0, 0, 0, 2, 0, 2, 0, 19, 0, 0, 0, 12, 0, 0, 0, 14, 0, 7, 0, 3, 0, 2, 0

Offset: 1

Des MacHale, Mar 02 2021

The smallest odd index n > 1 for which a(n) > 0 is 2187 = 3^7 (see A340521).

There exist even indices n that are not values taken by totient function phi (A002202) for which a(n) > 0. For example, John Bray has produced a group G that is the semidirect product 19:9 of order 3^2*19 = 171 such that |Aut(G)| = 1026 = 2*3^3*19.

			a(6) = 6, because there are six groups G with |Aut(G)| = 6. Four  cyclic groups: Aut(C_7) = Aut(C_9) = Aut(C_14) = Aut(C_18) ~~ C_6, and also Aut(C_2 x C_2) = Aut(S_3) ~~ S_3, where ~~ stands for “isomorphic to”. - _Bernard Schott_, Mar 02 2021
a(8) = 7, because there are seven groups G with |Aut(G)| = 8.

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.

Cf. A002202, A137315, A340521, A341823, A341824.

a(2) = 3, a(p) = 0 if p odd prime.

a(A002202(n)) > 0, since |Aut(C_n)| = phi(n).

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

Extensions