A365051 a(n) = |Aut^n(C_40)|: order of the group obtained by applying G -> Aut(G) n times to the cyclic group of order 40.

40, 16, 192, 1152, 4608, 18432

Offset

0,1

Comments

m = 40 is the next case after m = 32 where the sequence {Aut^n(C_m):n>=0} is not known to stabilize after some n. See A364904.

Links

Table of n, a(n) for n=0..5.

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_40) = C_2 X C_2 X C_4, so a(1) = 16;
Aut^2(C_40) = SmallGroup(192,1493), so a(2) = 192;
Aut^3(C_40) = SmallGroup(192,1493), so a(3) = 1152.

Prog

(GAP) A365051 := 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. A364904 ({Aut^n(C_32)}), A364917, A331921.

Sequence in context: A117831 A152143 A277874 * A033975 A033360 A370402

Adjacent sequences: A365048 A365049 A365050 * A365052 A365053 A365054

Keyword

nonn,hard,more

Author

Jianing Song, Aug 18 2023

Status

approved