

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.


2




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 wellknown 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



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

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



KEYWORD

nonn,hard,more


AUTHOR



STATUS

approved



