I tried to solve the puzzle, but I must be confused about something.
If you conjugate (a,b) with (c,d) I think you get (a, b c + d - a d), but most importantly the first element, a, is unchanged. So any automorphism of the kind the puzzle is describing must leave the a in x ↦ a x + b unchanged.
Suppose we try to find such an automorphism by brute force, and start by looking at what it does to the subgroup consisting of elements of the form x ↦ x + b (which it must preserve). All it can do is change b by permuting the elements of ℤ/8.
When I checked all 8! permutations of ℤ/8, the only nontrivial ones for which this is an automorphism are those that correspond to multiplication of b by 3, 5 or 7. But these are inner automorphisms, corresponding to conjugation by (3,0), (5,0) or (7,0).
I must be doing something wrong! Can you give me a hint as to where I’ve messed up?
Greg Egan /
npub12c…gd38p
2025-03-15 12:03:28
in reply to nevent1q…hetw
Author Public Key
npub12cuzzqzv8e8y7na77a9zv47mx2v6pec60qs5h7takzmv5p0mw0fsmgd38pPublished at
2025-03-15 12:03:28Event JSON
{
"id": "ffd0d6ac158e48e548287266fcbc2d95b6edeafb316095efa31da4541f0a5fb6",
"pubkey": "563821004c3e4e4f4fbef74a2657db3299a0e71a78214bf97db0b6ca05fb73d3",
"created_at": 1742040208,
"kind": 1,
"tags": [
[
"e",
"a4beab19db6cddb8e55e6c334995de4f3906f4005f2e8f008cd93d718915bfe5",
"",
"root",
"9a6a1a8eefd0b53d7e0c966ab35bd904151246f03b1be98fa0d2d0eeb4940320"
],
[
"proxy",
"https://mathstodon.xyz/@gregeganSF/114166347107808816",
"web"
],
[
"p",
"9a6a1a8eefd0b53d7e0c966ab35bd904151246f03b1be98fa0d2d0eeb4940320"
],
[
"proxy",
"https://mathstodon.xyz/users/gregeganSF/statuses/114166347107808816",
"activitypub"
],
[
"L",
"pink.momostr"
],
[
"l",
"pink.momostr.activitypub:https://mathstodon.xyz/users/gregeganSF/statuses/114166347107808816",
"pink.momostr"
],
[
"-"
]
],
"content": "I tried to solve the puzzle, but I must be confused about something.\n\nIf you conjugate (a,b) with (c,d) I think you get (a, b c + d - a d), but most importantly the first element, a, is unchanged. So any automorphism of the kind the puzzle is describing must leave the a in x ↦ a x + b unchanged.\n\nSuppose we try to find such an automorphism by brute force, and start by looking at what it does to the subgroup consisting of elements of the form x ↦ x + b (which it must preserve). All it can do is change b by permuting the elements of ℤ/8.\n\nWhen I checked all 8! permutations of ℤ/8, the only nontrivial ones for which this is an automorphism are those that correspond to multiplication of b by 3, 5 or 7. But these are inner automorphisms, corresponding to conjugation by (3,0), (5,0) or (7,0).\n\nI must be doing something wrong! Can you give me a hint as to where I’ve messed up?",
"sig": "573b96aecb6a1e018148d216f70d8a290e62d9da923caa3d4415b035c4e3d684fcd7634215e63c07e5f32435ef67ce194f3619c12c804ea07bca6386ac68c86d"
}