cp's OEIS Frontend

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

%I A341824 #35 Mar 11 2021 07:54:30
%S A341824 1,1,2,3,4,9,14,33
%N A341824 Number of groups of order 2^n which occur as Aut(G) for some finite group G.
%C A341824 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, ...
%C A341824 _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]
%H A341824 J. Flynn, D. MacHale, E. A. O'Brien and R. Sheehy, <a href="https://www.jstor.org/stable/20489479">Finite Groups whose Automorphism Groups are 2-groups</a>, Proc. Royal Irish Academy, 94A, (2) 1994, 137-145.
%F A341824 a(n) <= A000679(n). - _Des MacHale_, Mar 02 2021
%e A341824 a(5) = 9 because there are nine groups of order 32 which occur as automorphism groups of finite groups.
%e A341824 From _Bernard Schott_, Feb 26 2021: (Start)
%e A341824 Aut(C_15) = Aut(C_16) = Aut(C_20) = Aut(C_30) ~~ C_4 x C_2 where ~~ stands for "isomorphic to".
%e A341824 Aut(C_4 x C_2) = Aut(D_4) ~~ D_4 where D_4 is the dihedral group of the square.
%e A341824 Aut(C_24) ~~ C_2 x C_2 x C_2 = (C_2)^3.
%e A341824 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)
%Y A341824 Cf. A000679, A054397, A340521, A341823, A341825.
%K A341824 nonn,more
%O A341824 0,3
%A A341824 _Des MacHale_, Feb 26 2021
%E A341824 Offset modified by _Jianing Song_, Mar 02 2021