cp's OEIS Frontend

A364904 a(n) = |Aut^n(C_32)|: order of the group obtained by applying G -> Aut(G) n times to the cyclic group of order 32.

%I A364904 #15 Aug 19 2023 16:12:22
%S A364904 32,16,16,64,384,1536,6144
%N A364904 a(n) = |Aut^n(C_32)|: order of the group obtained by applying G -> Aut(G) n times to the cyclic group of order 32.
%C A364904 Also a(n) = |Aut^n(C_35)| for n >= 2, since Aut(Aut(C_32)) = Aut(Aut(C_35)) = C_2 X D_8.
%C A364904 The sequence {Aut^n(C_m):n>=0} is well-known for m <= 31. It is conjectured that |Aut^n(C_32)| tends to infinity as n goes to infinity.
%C A364904 This sequence appears in the table shown in the Math Overflow question "On the iterated automorphism groups of the cyclic groups" (see the Links section below).
%H A364904 G. Muller, <a href="https://mathoverflow.net/q/5635/34538">Does Aut(Aut(...Aut(G)...)) stabilize?</a>, MathOverflow (2009).
%H A364904 S. Palcoux, <a href="https://mathoverflow.net/q/351593/34538">On the iterated automorphism groups of the cyclic groups</a>, MathOverflow (2020).
%e A364904 Aut(C_32) = C_2 X C_8, so a(1) = 16;
%e A364904 Aut^2(C_32) = C_2 X D_8, so a(2) = 16;
%e A364904 Aut^3(C_32) = SmallGroup(64,138), so a(3) = 64;
%e A364904 Aut^4(C_32) = SmallGroup(384,17948), so a(4) = 384.
%o A364904 (GAP) A364904 := function(n)
%o A364904 local G, i, L;
%o A364904 G := CyclicGroup(32);
%o A364904 for i in [1..n] do
%o A364904 G := AutomorphismGroup(G);
%o A364904 if i = n then return break; fi;
%o A364904 L := DirectFactorsOfGroup(G);
%o A364904 if List(L, x->IdGroupsAvailable(Size(x))) = List(L, x->true) then
%o A364904 L := List(L, x->IdGroup(x));
%o A364904 G := DirectProduct(List(L, x->SmallGroup(x))); # It's more efficient to operate on abstract groups when the abstract structure is available
%o A364904 fi; od;
%o A364904 return Size(G);
%o A364904 end;
%Y A364904 Cf. A365051 ({Aut^n(C_40)}), A364917, A331921.
%K A364904 nonn,hard,more
%O A364904 0,1
%A A364904 _Jianing Song_, Aug 12 2023