OFFSET
1,8
COMMENTS
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; divisors of 16, 20, 26, 30, 34, 46, 50, 58, 62, 74, 94, 118, 134, 146, 168, 172, 178, 196, 218, 258, 264, 294, 346, 394, 456, 458, 648, 686, 694 or 914.
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 (order = 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 (order = 1);
- 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 (order = 1);
- C_13 or C_26 -> C_8 or C_12 -> C_2 X C_2 -> S_3 (order = 6);
- 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 (order = 8);
- 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 (order = 12);
- C_24 -> C_2 X C_2 X C_2 -> PSL(2,7) -> PGL(2,7) (order = 336);
- 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 (order = 144);
- 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 (order = 144);
- 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) (order = 336);
- 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) (order = 672);
- 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) (order = 165888);
- 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).(order = 672);
- C_168 -> C_2 X C_2 X C_2 X C_6 -> C_2 X A_8 -> S_8 (order = 40320);
- C_264 -> C_2 X C_2 X C_2 X C_10 -> C_4 X A_8 -> C_2 X S_8 (order = 80640);
- 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 (order = 80640);
- 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) (order = 165888);
- 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) (order = 8064).
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)? (order = 2322432);
- 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)? (order = 278691840).
Note that a(35) = a(32) (although the value of each is unknown), since Aut(Aut(C_32)) = Aut(Aut(C_35)) = C_2 X D_8.
LINKS
G. Muller, Does Aut(Aut(...Aut(G)...)) stabilize?, MathOverflow (2009).
S. Palcoux, On the iterated automorphism groups of the cyclic groups, MathOverflow (2020).
EXAMPLE
For n = 24, we have G_0 = C_24, G_1 = C_2 X C_2 X C_2, G_2 = PSL(2,7) and G_k = PGL(2,7) for all k >= 3, hence a(24) = |PGL(2,7)| = 336.
CROSSREFS
KEYWORD
nonn,hard,more
AUTHOR
Jianing Song, Aug 12 2023
STATUS
approved