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I think I’ve found a class-preserving outer automorphism of Aff(ℤ/8)!
You map:
x ↦ a x + b
to:
x ↦ a x + (b + D(a))
where
D(1) = D(3) = 0
D(5) = D(7) = 4
I don’t think this is unique. But I’m still searching through the (4!)^8 choices of permutations of the odd and even parts of ℤ/8, for each value of a.
Conjugation preserves the parity of b, so rather than there being 8! choices of permutations of ℤ/8 for each value of a, there are (4!)^2.
Greg Egan /
npub1qq…j5nhv
2025-03-16 05:20:06
in reply to nevent1q…qff8
Author Public Key
npub1qqm7nu2qf25xd3mw6y6cyp4v2wr7ktf43ymp5wqz4u8jvz76ymtsjj5nhvSeen on
wss://relay.mostr.pubPublished at
2025-03-16 05:20:06Event JSON
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"content": "nostr:nprofile1qy2hwumn8ghj7un9d3shjtnddaehgu3wwp6kyqpqknzsux7p6lzwzdedp3m8c3c92z0swzc0xyy5glvse58txj5e9ztqaufa4k nostr:nprofile1qy2hwumn8ghj7un9d3shjtnddaehgu3wwp6kyqpqqmshcslmwan3erj3ulczepe8q8a8xdyfkndnk8fudgtg8ccwg6zqlsl9jz \n\nI think I’ve found a class-preserving outer automorphism of Aff(ℤ/8)!\n\nYou map:\n\nx ↦ a x + b\n\nto:\n\nx ↦ a x + (b + D(a))\n\nwhere\n\nD(1) = D(3) = 0\nD(5) = D(7) = 4\n\nI don’t think this is unique. But I’m still searching through the (4!)^8 choices of permutations of the odd and even parts of ℤ/8, for each value of a.\n\nConjugation preserves the parity of b, so rather than there being 8! choices of permutations of ℤ/8 for each value of a, there are (4!)^2.",
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