A364904 - OEIS

%I #15 Aug 19 2023 16:12:22

%S 32,16,16,64,384,1536,6144

%N 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 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 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 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 G. Muller, <a href="https://mathoverflow.net/q/5635/34538">Does Aut(Aut(...Aut(G)...)) stabilize?</a>, MathOverflow (2009).

%H S. Palcoux, <a href="https://mathoverflow.net/q/351593/34538">On the iterated automorphism groups of the cyclic groups</a>, MathOverflow (2020).

%e Aut(C_32) = C_2 X C_8, so a(1) = 16;

%e Aut^2(C_32) = C_2 X D_8, so a(2) = 16;

%e Aut^3(C_32) = SmallGroup(64,138), so a(3) = 64;

%e Aut^4(C_32) = SmallGroup(384,17948), so a(4) = 384.

%o (GAP) A364904 := function(n)

%o local G, i, L;

%o G := CyclicGroup(32);

%o for i in [1..n] do

%o G := AutomorphismGroup(G);

%o if i = n then return break; fi;

%o L := DirectFactorsOfGroup(G);

%o if List(L, x->IdGroupsAvailable(Size(x))) = List(L, x->true) then

%o L := List(L, x->IdGroup(x));

%o G := DirectProduct(List(L, x->SmallGroup(x))); # It's more efficient to operate on abstract groups when the abstract structure is available

%o fi; od;

%o return Size(G);

%o end;

%Y Cf. A365051 ({Aut^n(C_40)}), A364917, A331921.

%K nonn,hard,more

%O 0,1

%A _Jianing Song_, Aug 12 2023