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
Examples
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.
Links
- G. Muller, Does Aut(Aut(...Aut(G)...)) stabilize?, MathOverflow (2009).
- S. Palcoux, On the iterated automorphism groups of the cyclic groups, MathOverflow (2020).
Programs
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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;
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