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.

32, 16, 16, 64, 384, 1536, 6144

Offset

0,1

Comments

Also a(n) = |Aut^n(C_35)| for n >= 2, since Aut(Aut(C_32)) = Aut(Aut(C_35)) = C_2 X D_8.

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.

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).

Links

Table of n, a(n) for n=0..6.

G. Muller, Does Aut(Aut(...Aut(G)...)) stabilize?, MathOverflow (2009).

S. Palcoux, On the iterated automorphism groups of the cyclic groups, MathOverflow (2020).

Example

Aut(C_32) = C_2 X C_8, so a(1) = 16;
Aut^2(C_32) = C_2 X D_8, so a(2) = 16;
Aut^3(C_32) = SmallGroup(64,138), so a(3) = 64;
Aut^4(C_32) = SmallGroup(384,17948), so a(4) = 384.

Prog

(GAP) A364904 := function(n)
local G, i, L;
G := CyclicGroup(32);
for i in [1..n] do
G := AutomorphismGroup(G);
if i = n then return break; fi;
L := DirectFactorsOfGroup(G);
if List(L, x->IdGroupsAvailable(Size(x))) = List(L, x->true) then
L := List(L, x->IdGroup(x));
G := DirectProduct(List(L, x->SmallGroup(x))); # It's more efficient to operate on abstract groups when the abstract structure is available
fi; od;
return Size(G);
end;

Crossrefs

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

Sequence in context: A070621 A234967 A033352 * A140387 A023094 A087502

Adjacent sequences: A364901 A364902 A364903 * A364905 A364906 A364907

Keyword

nonn,hard,more

Author

Jianing Song, Aug 12 2023

Status

approved