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A364904
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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.
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2
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OFFSET
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0,1
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COMMENTS
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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).
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LINKS
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EXAMPLE
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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.
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PROG
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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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CROSSREFS
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KEYWORD
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nonn,hard,more
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AUTHOR
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STATUS
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approved
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