A365051 - OEIS

%I #10 Aug 19 2023 16:12:37

%S 40,16,192,1152,4608,18432

%N 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.

%C 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.

%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_40) = C_2 X C_2 X C_4, so a(1) = 16;

%e Aut^2(C_40) = SmallGroup(192,1493), so a(2) = 192;

%e Aut^3(C_40) = SmallGroup(192,1493), so a(3) = 1152.

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

%K nonn,hard,more

%O 0,1

%A _Jianing Song_, Aug 18 2023