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.

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

Offset: 1

Author

Sébastien Palcoux, Feb 01 2020

Keywords

• nonn
• more

Comments

a(n) is the smallest m >= 0 such that Aut^{m+r}(Cn) is isomorphic to Aut^m(Cn) for some r > 0.

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).

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.

Question: What is a(32)? (we just know that a(32)>=8)

Conjecture: a(n) != -1 for all n.

Question: Is there n such that the sequence Aut^m(Cn) reaches a loop of length>1?

From Jianing Song, Aug 13 2023 and edited on 24 Feb 2025: (Start)

It can be checked that a(n) != -1 for the following numbers:

- 3^k and 2*3^k for all k >= 0;

- 2*3^k+1 and 2*(2*3^k+1) for all k >= 0, where 2*3^k+1 is a prime;

- 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;

- divisors of 16, 20, 30, 50, 168, 172, 196, 258, 264, 294, 456, 648, or 686.

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_{3^k}, C_{2*3^k} -> C_{2*3^(k-1)} -> ... -> C_2 -> C_1, k >= 1;

- 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_13 or C_26 -> C_8 or C_12 -> C_2 X C_2 -> S_3;

- 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_24 -> C_2 X C_2 X C_2 -> PSL(2,7) -> PGL(2,7);

- 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_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_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_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_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_168 -> C_2 X C_2 X C_2 X C_6 -> C_2 X A_8 -> S_8;

- 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_264 -> C_2 X C_2 X C_2 X C_10 -> C_4 X A_8 -> C_2 X S_8;

- 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_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_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_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_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).

The following two sequences are conjectured to be correct and to stabilize at the last term:

- 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_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)

Examples

			Aut(C1)=C1 so a(1)=0.
Aut(C2)=C1 so a(2)=1.
Aut(C8)=C2xC2, Aut(C2xC2)=S3, Aut(S3)=S3, so a(8)=2 (whereas A003434(8)=3).
Aut(C15)=C2xC4, Aut(C2xC4)=D8, Aut(D8)=D8, so a(15)=2 (whereas A185816(15)=3).
		

Links

• A. P. Goucher, Iterated automorphism groups of finite groups, MathOverflow (2018).
• G. Muller, Does Aut(Aut(...Aut(G)...)) stabilize?, MathOverflow (2009).
• S. Palcoux, On the iterated automorphism groups of the cyclic groups, MathOverflow (2020).
• Wikipedia, Multiplicative group of integers modulo n.

Crossrefs

Cf. A000010, A003434, A185816, A117729, A179401.

Programs

GAP

gap> LoadPackage("sonata");
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;
gap> L;
[ 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 ]
gap> SG;
[ [ 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 ] ]
gap> Set(SG);
[ [ 1, 1 ], [ 6, 1 ], [ 8, 3 ], [ 12, 4 ], [ 336, 208 ] ]
# It is the list of IdGroup for C1, S3, D8, D12 and PGL(2,7).
# 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.

Extensions

Escape clause added by Jianing Song, Aug 13 2023