A341825 -id:A341825 - cp's OEIS frontend

A341823 Number of finite groups G with |Aut(G)| = 2^n.

2, 3, 4, 7, 11, 19, 34, 70

Offset: 0

Des MacHale, Feb 20 2021

This sequence is infinite, but the amount of computation needed to consider the large number of groups of order 2^8 suggests it may be hard to find the next term.

			a(3) = 7, because there are seven finite groups G with |Aut(G)| = 8. Four cyclic groups: Aut(C_15) = Aut(C_16) = Aut(C_20) = Aut(C_30) ~~ C_4 x C_2, also Aut(C_4 x C_2) = Aut(D_4) ~~ D_4, with D_4 is the dihedral group of the square, finally Aut(C_24) ~~ C_2 x C_2 x C_2 = (C_2)^3 where ~~ stands for “isomorphic to". - _Bernard Schott_, Mar 04 2021

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. A341824, A341825.

Subsequence of A340521.

Offset modified by Bernard Schott, Mar 04 2021

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

Original entry on oeis.org

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

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A341823 Number of finite groups G with |Aut(G)| = 2^n.

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