cp's OEIS Frontend

A331921 Number of iterations of the map G -> Aut(G) on the cyclic group of order n needed to reach stability; or -1 if the iteration never stabilizes.

%I A331921 #71 Feb 25 2025 10:35:48
%S A331921 0,1,2,2,3,2,3,2,3,3,4,2,3,3,2,2,3,3,4,2,2,4,5,3,3,3,4,2,3,2,3
%N A331921 Number of iterations of the map G -> Aut(G) on the cyclic group of order n needed to reach stability; or -1 if the iteration never stabilizes.
%C A331921 a(n) is the smallest m >= 0 such that Aut^{m+r}(Cn) is isomorphic to Aut^m(Cn) for some r > 0.
%C A331921 This sequence shares the first 7 terms with A003434 but not beyond, because Aut(Cn) has order phi(n) (see A000010) but need not be cyclic. It also shares the first 14 terms with A185816 (not beyond).
%C A331921 For n<32, G=Aut^{m+1}(Cn) is isomorphic to Aut^m(Cn) iff G is in {C1,S3,D8,D12,PGL(2,7)}. This is established by the GAP computation below.
%C A331921 Question: What is a(32)? (we just know that a(32)>=8)
%C A331921 Conjecture: a(n) != -1 for all n.
%C A331921 Question: Is there n such that the sequence Aut^m(Cn) reaches a loop of length>1?
%C A331921 From _Jianing Song_, Aug 13 2023 and edited on 24 Feb 2025: (Start)
%C A331921 It can be checked that a(n) != -1 for the following numbers:
%C A331921  - 3^k and 2*3^k for all k >= 0;
%C A331921  - 2*3^k+1 and 2*(2*3^k+1) for all k >= 0, where 2*3^k+1 is a prime;
%C A331921  - p and 2*p for p = 13, 17, 23, 29, 31, 37, 47, 59, 67, 73, 89, 109, 173, 179, 197, 229, 347, 359, 457, 719, 1439, or 2879;
%C A331921  - divisors of 16, 20, 30, 50, 168, 172, 196, 258, 264, 294, 456, 648, or 686.
%C A331921 The sequences of iterations are listed as follows (D_{2n} = dihedral group of order 2*n, S_n = symmetric group over set of size n, A_n = alternating group over set of size n):
%C A331921  - C_{3^k}, C_{2*3^k} -> C_{2*3^(k-1)} -> ... -> C_2 -> C_1, k >= 1;
%C A331921  - C_{2*3^k+1} or C_{2*(2*3^k+1)} -> C_{2*3^k} -> ... -> C_2 -> C_1, k >= 0, 2*3^k+1 is prime;
%C A331921  - C_13 or C_26 -> C_8 or C_12 -> C_2 X C_2 -> S_3;
%C A331921  - C_17, C_25, C_31, C_34, C_50, or C_62 -> C_15, C_16, C_20, or C_30 -> C_2 X C_4 -> D_8;
%C A331921  - C_24 -> C_2 X C_2 X C_2 -> PSL(2,7) -> PGL(2,7);
%C A331921  - C_47 or C_94 -> C_23 or C_46 -> C_11 or C_22 -> C_5 or C_10 -> C_4 -> C_2 -> C_1;
%C A331921  - C_59 or C_118 -> C_29, C_37, C_43, C_49, C_58, C_74, C_86, or C_98 -> C_21, C_28, C_36, or C_42 -> C_2 X C_6 -> D_12;
%C A331921  - C_67 or C_134 -> C_33, C_44, or C_66 -> C_2 X C_10 -> C_4 X S_3 -> C_2 X D_12 -> S_3 X S_4;
%C A331921  - C_73 or C_146 -> C_56, C_72, or C_84 -> C_2 X C_2 X C_6 -> C_2 X PSL(2,7) -> PGL(2,7);
%C A331921  - C_109 or C_218 -> C_57, C_76, C_108, or C_114 -> C_2 X C_18 -> C_6 X S_3 -> C_2 X D_12 -> S_3 X S_4;
%C A331921  - C_168 -> C_2 X C_2 X C_2 X C_6 -> C_2 X A_8 -> S_8;
%C A331921  - C_229 or C_458 -> C_152, C_216 or C_228 -> C_2 X C_2 X C_18 -> C_6 X PSL(2,7) -> C_2 X PGL(2,7);
%C A331921  - C_264 -> C_2 X C_2 X C_2 X C_10 -> C_4 X A_8 -> C_2 X S_8;
%C A331921  - C_324 -> C_2 X C_54 -> C_18 X S_3 -> C_6 X D_12 -> C_2 X D_12 X S_4 -> S_3 X S_4 X SmallGroup(96,227) -> S_3 X S_4 X SmallGroup(576,8654) -> S_3 X S_4 X SmallGroup(1152,157849);
%C A331921  - C_347 or C_694 -> C_173, C_197, C_343, C_346, C_394 or C_686 -> C_129, C_147, C_172, C_196, C_258, or C_294 -> C_2 X C_42 -> C_6 X D_12 -> C_2 X D_12 X S_4 -> S_3 X S_4 X SmallGroup(96,227) -> S_3 X S_4 X SmallGroup(576,8654) -> S_3 X S_4 X SmallGroup(1152,157849);
%C A331921  - C_457 or C_914 -> C_456 -> C_2 X C_2 X C_2 X C_18 -> C_6 X A_8 -> C_2 X S_8;
%C A331921  - C_648 -> C_2 X C_2 X C_54 -> C_18 X PSL(2,7) -> C_6 X PGL(2,7) -> C_2 X C_2 X PGL(2,7) -> S_4 X PGL(2,7);
%C A331921  - C_2879 or C_5758 -> C_1439 or C_2878 -> C_719 or C_1438 -> C_359 or C_718 -> C_179 or C_358 -> C_89 or C_178 -> C_88 or C_132 -> C_2 X C_2 X C_10 -> C_4 X PSL(2,7) -> C_2 X PGL(2,7).
%C A331921 The following two sequences are conjectured to be correct and to stabilize at the last term:
%C A331921  - C_344, C_392, C_516, or C_588 -> C_2 X C_2 X C_42 -> C_2 X C_6 X PSL(2,7) - > D_12 X PGL(2,7) -> C_2 X D_12 X PGL(2,7) -> S_3 X PGL(2,7) X SmallGroup(96,227) -> S_3 X PGL(2,7) X SmallGroup(576,8654)? -> S_3 X PGL(2,7) X SmallGroup(1152,157849)?
%C A331921  - C_1033 or C_2066 -> C_1032 or C_1176 -> C_2 X C_2 X C_2 X C_42 -> C_2 X C_6 X A_8 - > D_12 X S_8 -> C_2 X D_12 X S_8? -> S_3 X S_8 X SmallGroup(96,227)? -> S_3 X S_8 X SmallGroup(576,8654)? -> S_3 X S_8 X SmallGroup(1152,157849)? (End)
%H A331921 A. P. Goucher, <a href="https://mathoverflow.net/q/294229/34538">Iterated automorphism groups of finite groups</a>, MathOverflow (2018).
%H A331921 G. Muller, <a href="https://mathoverflow.net/q/5635/34538">Does Aut(Aut(...Aut(G)...)) stabilize?</a>, MathOverflow (2009).
%H A331921 S. Palcoux, <a href="https://mathoverflow.net/q/351593/34538">On the iterated automorphism groups of the cyclic groups</a>, MathOverflow (2020).
%H A331921 Wikipedia, <a href="https://en.wikipedia.org/wiki/Multiplicative_group_of_integers_modulo_n">Multiplicative group of integers modulo n</a>.
%e A331921 Aut(C1)=C1 so a(1)=0.
%e A331921 Aut(C2)=C1 so a(2)=1.
%e A331921 Aut(C8)=C2xC2, Aut(C2xC2)=S3, Aut(S3)=S3, so a(8)=2 (whereas A003434(8)=3).
%e A331921 Aut(C15)=C2xC4, Aut(C2xC4)=D8, Aut(D8)=D8, so a(15)=2 (whereas A185816(15)=3).
%o A331921 (GAP)
%o A331921 gap> LoadPackage("sonata");
%o A331921 gap> L:=[];; SG:=[];; for n in [1..31] do a:=0; C:=CyclicGroup(n); A:=AutomorphismGroup(C); while Order(C)<>Order(A) or not IsIsomorphicGroup(A,C) do a:=a+1; C:=A; A:=AutomorphismGroup(A); od; Add(L,a); Add(SG,IdGroup(A)); od;
%o A331921 gap> L;
%o A331921 [ 0, 1, 2, 2, 3, 2, 3, 2, 3, 3, 4, 2, 3, 3, 2, 2, 3, 3, 4, 2, 2, 4, 5, 3, 3, 3, 4, 2, 3, 2, 3 ]
%o A331921 gap> SG;
%o A331921 [ [ 1, 1 ], [ 1, 1 ], [ 1, 1 ], [ 1, 1 ], [ 1, 1 ], [ 1, 1 ], [ 1, 1 ], [ 6, 1 ], [ 1, 1 ], [ 1, 1 ], [ 1, 1 ], [ 6, 1 ], [ 6, 1 ], [ 1, 1 ], [ 8, 3 ], [ 8, 3 ], [ 8, 3 ], [ 1, 1 ], [ 1, 1 ], [ 8, 3 ], [ 12, 4 ], [ 1, 1 ], [ 1, 1 ], [ 336, 208 ], [ 8, 3 ], [ 6, 1 ], [ 1, 1 ], [ 12, 4 ], [ 12, 4 ], [ 8, 3 ], [ 8, 3 ] ]
%o A331921 gap> Set(SG);
%o A331921 [ [ 1, 1 ], [ 6, 1 ], [ 8, 3 ], [ 12, 4 ], [ 336, 208 ] ]
%o A331921 # It is the list of IdGroup for C1, S3, D8, D12 and PGL(2,7).
%o A331921 # The above program works well for n<32. Beyond, it will work as long as there is no loop of length>1 and a(n) finite, which (for small n) is very likely (the opposite would be a breakthrough), otherwise it will just not end. Moreover, if Order(A) is too big then IdGroup(A) will not work, because the SmallGroup library of GAP is finite.
%Y A331921 Cf. A000010, A003434, A185816, A117729, A179401.
%K A331921 nonn,more
%O A331921 1,3
%A A331921 _Sébastien Palcoux_, Feb 01 2020
%E A331921 Escape clause added by _Jianing Song_, Aug 13 2023